Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 88.2%

Time: 16.9s

Alternatives: 16

Speedup: 24.8×

• 27.7% 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: 55.2% 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: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1320:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot {\left(\sqrt[3]{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1320.0)
    (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t_m (pow (sin k) 2.0)))))
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* (tan k) (pow (cbrt (+ 2.0 (pow (/ k t_m) 2.0))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1320.0) {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * pow(cbrt((2.0 + pow((k / t_m), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 1320.0) {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * Math.pow(Math.cbrt((2.0 + Math.pow((k / t_m), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1320.0)
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * (cbrt(Float64(2.0 + (Float64(k / t_m) ^ 2.0))) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1320.0], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[Power[N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1320

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

      1. add-cube-cbrt 57.6%

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

      2. pow3 57.6%

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

      3. associate-/r* 61.0%

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

      4. *-commutative 61.0%

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

      5. cbrt-prod 60.9%

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

      6. associate-/r* 57.6%

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

      7. cbrt-div 58.5%

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

      8. rem-cbrt-cube 64.7%

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

      9. cbrt-prod 73.2%

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

      10. pow2 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. pow1 73.2%

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

      2. associate-+r+ 73.2%

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

      3. metadata-eval 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. unpow1 73.2%

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

   10. Simplified 73.2%

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

   11. Taylor expanded in k around inf 63.1%

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

       1. times-frac 64.6%

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

       2. unpow2 64.6%

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

       3. unpow2 64.6%

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

       4. times-frac 74.6%

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

       5. *-rgt-identity 74.6%

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

       6. associate-*r/ 74.6%

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

       7. pow-base-1 74.6%

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

       8. *-rgt-identity 74.6%

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

       9. associate-*r/ 74.6%

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

       10. pow-base-1 74.6%

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

       11. unpow2 74.6%

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

       12. pow-base-1 74.6%

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

       13. associate-*r/ 74.6%

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

       14. *-rgt-identity 74.6%

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

   13. Simplified 74.6%

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

   if 1320 < t

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

      1. add-cube-cbrt 68.1%

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

      2. pow3 68.1%

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

      3. associate-/r* 75.3%

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

      4. *-commutative 75.3%

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

      5. cbrt-prod 75.3%

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

      6. associate-/r* 68.1%

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

      7. cbrt-div 68.0%

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

      8. rem-cbrt-cube 71.8%

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

      9. cbrt-prod 95.8%

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

      10. pow2 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. associate-+r+ 95.8%

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

      2. metadata-eval 95.8%

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

      3. add-cube-cbrt 95.8%

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

      4. pow3 95.9%

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

   8. Applied egg-rr 95.9%

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

Alternative 2: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 15000:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 15000.0)
    (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t_m (pow (sin k) 2.0)))))
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 15000.0) {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (2.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 15000.0) {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 15000.0)
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 15000.0], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 15000

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

      1. add-cube-cbrt 57.6%

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

      2. pow3 57.6%

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

      3. associate-/r* 61.0%

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

      4. *-commutative 61.0%

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

      5. cbrt-prod 60.9%

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

      6. associate-/r* 57.6%

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

      7. cbrt-div 58.5%

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

      8. rem-cbrt-cube 64.7%

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

      9. cbrt-prod 73.2%

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

      10. pow2 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. pow1 73.2%

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

      2. associate-+r+ 73.2%

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

      3. metadata-eval 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. unpow1 73.2%

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

   10. Simplified 73.2%

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

   11. Taylor expanded in k around inf 63.1%

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

       1. times-frac 64.6%

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

       2. unpow2 64.6%

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

       3. unpow2 64.6%

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

       4. times-frac 74.6%

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

       5. *-rgt-identity 74.6%

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

       6. associate-*r/ 74.6%

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

       7. pow-base-1 74.6%

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

       8. *-rgt-identity 74.6%

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

       9. associate-*r/ 74.6%

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

       10. pow-base-1 74.6%

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

       11. unpow2 74.6%

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

       12. pow-base-1 74.6%

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

       13. associate-*r/ 74.6%

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

       14. *-rgt-identity 74.6%

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

   13. Simplified 74.6%

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

   if 15000 < t

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

      1. add-cube-cbrt 68.1%

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

      2. pow3 68.1%

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

      3. associate-/r* 75.3%

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

      4. *-commutative 75.3%

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

      5. cbrt-prod 75.3%

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

      6. associate-/r* 68.1%

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

      7. cbrt-div 68.0%

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

      8. rem-cbrt-cube 71.8%

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

      9. cbrt-prod 95.8%

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

      10. pow2 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. pow1 95.8%

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

      2. associate-+r+ 95.8%

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

      3. metadata-eval 95.8%

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

   8. Applied egg-rr 95.8%

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

      1. unpow1 95.8%

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

   10. Simplified 95.8%

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

Alternative 3: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e+38)
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* 2.0 (/ (sin k) (cos k)))))
    (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e+38) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * (sin(k) / cos(k))));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 2.2e+38) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * (Math.sin(k) / Math.cos(k))));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e+38)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * Float64(sin(k) / cos(k)))));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2.2e+38], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000006e38

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

      3. associate-/r* 68.4%

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

      4. *-commutative 68.4%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 65.1%

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

      7. cbrt-div 65.9%

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

      8. rem-cbrt-cube 70.9%

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

      9. cbrt-prod 83.3%

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

      10. pow2 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in t around inf 80.4%

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

   if 2.20000000000000006e38 < k

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

      1. add-cube-cbrt 39.8%

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

      2. pow3 39.8%

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

      3. associate-/r* 47.6%

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

      4. *-commutative 47.6%

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

      5. cbrt-prod 47.6%

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

      6. associate-/r* 39.9%

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

      7. cbrt-div 39.8%

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

      8. rem-cbrt-cube 48.0%

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

      9. cbrt-prod 59.1%

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

      10. pow2 59.1%

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

   6. Applied egg-rr 59.1%

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

      1. pow1 59.1%

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

      2. associate-+r+ 59.1%

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

      3. metadata-eval 59.1%

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

   8. Applied egg-rr 59.1%

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

      1. unpow1 59.1%

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

   10. Simplified 59.1%

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

   11. Taylor expanded in k around inf 62.7%

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

       1. times-frac 62.9%

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

       2. unpow2 62.9%

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

       3. unpow2 62.9%

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

       4. times-frac 89.6%

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

       5. *-rgt-identity 89.6%

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

       6. associate-*r/ 89.6%

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

       7. pow-base-1 89.6%

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

       8. *-rgt-identity 89.6%

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

       9. associate-*r/ 89.5%

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

       10. pow-base-1 89.5%

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

       11. unpow2 89.5%

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

       12. pow-base-1 89.5%

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

       13. associate-*r/ 89.6%

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

       14. *-rgt-identity 89.6%

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

   13. Simplified 89.6%

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

Alternative 4: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e+37)
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* 2.0 (tan k))))
    (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e+37) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * tan(k)));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 1.7e+37) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * Math.tan(k)));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.7e+37)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * tan(k))));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 1.7e+37], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000003e37

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

      3. associate-/r* 68.4%

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

      4. *-commutative 68.4%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 65.1%

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

      7. cbrt-div 65.9%

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

      8. rem-cbrt-cube 70.9%

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

      9. cbrt-prod 83.3%

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

      10. pow2 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in k around 0 80.4%

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

   if 1.70000000000000003e37 < k

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

      1. add-cube-cbrt 39.8%

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

      2. pow3 39.8%

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

      3. associate-/r* 47.6%

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

      4. *-commutative 47.6%

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

      5. cbrt-prod 47.6%

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

      6. associate-/r* 39.9%

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

      7. cbrt-div 39.8%

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

      8. rem-cbrt-cube 48.0%

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

      9. cbrt-prod 59.1%

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

      10. pow2 59.1%

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

   6. Applied egg-rr 59.1%

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

      1. pow1 59.1%

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

      2. associate-+r+ 59.1%

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

      3. metadata-eval 59.1%

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

   8. Applied egg-rr 59.1%

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

      1. unpow1 59.1%

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

   10. Simplified 59.1%

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

   11. Taylor expanded in k around inf 62.7%

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

       1. times-frac 62.9%

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

       2. unpow2 62.9%

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

       3. unpow2 62.9%

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

       4. times-frac 89.6%

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

       5. *-rgt-identity 89.6%

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

       6. associate-*r/ 89.6%

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

       7. pow-base-1 89.6%

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

       8. *-rgt-identity 89.6%

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

       9. associate-*r/ 89.5%

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

       10. pow-base-1 89.5%

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

       11. unpow2 89.5%

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

       12. pow-base-1 89.5%

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

       13. associate-*r/ 89.6%

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

       14. *-rgt-identity 89.6%

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

   13. Simplified 89.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.04:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.04)
    (/
     2.0
     (* (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0) (* 2.0 k)))
    (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.04) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 0.04) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.04)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 0.04], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0400000000000000008

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.9%

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

      2. pow3 65.9%

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

      3. associate-/r* 69.3%

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

      4. *-commutative 69.3%

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

      5. cbrt-prod 69.2%

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

      6. associate-/r* 65.9%

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

      7. cbrt-div 66.8%

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

      8. rem-cbrt-cube 71.9%

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

      9. cbrt-prod 83.7%

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

      10. pow2 83.7%

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

   6. Applied egg-rr 83.7%

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

   7. Taylor expanded in k around 0 78.2%

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

   if 0.0400000000000000008 < k

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

      1. add-cube-cbrt 40.2%

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

      2. pow3 40.2%

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

      3. associate-/r* 47.2%

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

      4. *-commutative 47.2%

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

      5. cbrt-prod 47.1%

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

      6. associate-/r* 40.2%

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

      7. cbrt-div 40.1%

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

      8. rem-cbrt-cube 47.4%

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

      9. cbrt-prod 60.4%

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

      10. pow2 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. pow1 60.4%

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

      2. associate-+r+ 60.4%

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

      3. metadata-eval 60.4%

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

   8. Applied egg-rr 60.4%

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

      1. unpow1 60.4%

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

   10. Simplified 60.4%

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

   11. Taylor expanded in k around inf 62.2%

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

       1. times-frac 62.4%

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

       2. unpow2 62.4%

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

       3. unpow2 62.4%

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

       4. times-frac 85.9%

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

       5. *-rgt-identity 85.9%

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

       6. associate-*r/ 85.9%

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

       7. pow-base-1 85.9%

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

       8. *-rgt-identity 85.9%

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

       9. associate-*r/ 85.8%

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

       10. pow-base-1 85.8%

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

       11. unpow2 85.8%

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

       12. pow-base-1 85.8%

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

       13. associate-*r/ 85.9%

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

       14. *-rgt-identity 85.9%

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

   13. Simplified 85.9%

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

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t\_m}^{-1.5}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.55e-5)
    (pow (* (/ l k) (pow t_m -1.5)) 2.0)
    (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.55e-5) {
		tmp = pow(((l / k) * pow(t_m, -1.5)), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.55d-5) then
        tmp = ((l / k) * (t_m ** (-1.5d0))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * (cos(k) / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 1.55e-5) {
		tmp = Math.pow(((l / k) * Math.pow(t_m, -1.5)), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.55e-5:
		tmp = math.pow(((l / k) * math.pow(t_m, -1.5)), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * (math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.55e-5)
		tmp = Float64(Float64(l / k) * (t_m ^ -1.5)) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.55e-5)
		tmp = ((l / k) * (t_m ^ -1.5)) ^ 2.0;
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * (cos(k) / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 1.55e-5], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000007e-5

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-sqr-sqrt 46.6%

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

   6. Applied egg-rr 45.9%

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

      1. unpow2 45.9%

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

      2. associate-/l* 45.9%

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

      3. associate-*l* 41.7%

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

   8. Simplified 41.7%

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

   9. Taylor expanded in k around 0 42.4%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 42.4%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 42.4%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 42.4%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. pow1 42.4%

          \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-*r/ 42.4%

          \[\leadsto {\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       3. pow1/2 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       4. pow1/2 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       5. pow-prod-down 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       6. metadata-eval 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   13. Applied egg-rr 42.4%

       \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]
   14. Step-by-step derivation

       1. unpow1 42.4%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. pow-base-1 42.4%

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

       3. associate-*r/ 42.4%

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

       4. *-rgt-identity 42.4%

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

   15. Simplified 42.4%

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

       1. *-un-lft-identity 42.4%

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

       2. pow1/2 44.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)\right)}^{2} \]

       3. pow-flip 44.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)\right)}^{2} \]

       4. pow-pow 31.7%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)\right)}^{2} \]

       5. metadata-eval 31.7%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)\right)}^{2} \]

       6. metadata-eval 31.7%

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

   17. Applied egg-rr 31.7%

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

       1. *-lft-identity 31.7%

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

   19. Simplified 31.7%

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

   if 1.55000000000000007e-5 < k

   1. Initial program 40.2%

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

   3. Simplified 40.2%

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

      1. add-cube-cbrt 40.2%

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

      2. pow3 40.2%

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

      3. associate-/r* 47.2%

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

      4. *-commutative 47.2%

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

      5. cbrt-prod 47.1%

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

      6. associate-/r* 40.2%

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

      7. cbrt-div 40.1%

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

      8. rem-cbrt-cube 47.4%

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

      9. cbrt-prod 60.4%

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

      10. pow2 60.4%

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

   6. Applied egg-rr 60.4%

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

      1. pow1 60.4%

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

      2. associate-+r+ 60.4%

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

      3. metadata-eval 60.4%

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

   8. Applied egg-rr 60.4%

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

      1. unpow1 60.4%

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

   10. Simplified 60.4%

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

   11. Taylor expanded in k around inf 62.2%

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

       1. times-frac 62.4%

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

       2. unpow2 62.4%

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

       3. unpow2 62.4%

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

       4. times-frac 85.9%

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

       5. *-rgt-identity 85.9%

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

       6. associate-*r/ 85.9%

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

       7. pow-base-1 85.9%

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

       8. *-rgt-identity 85.9%

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

       9. associate-*r/ 85.8%

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

       10. pow-base-1 85.8%

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

       11. unpow2 85.8%

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

       12. pow-base-1 85.8%

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

       13. associate-*r/ 85.9%

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

       14. *-rgt-identity 85.9%

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

   13. Simplified 85.9%

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

Alternative 7: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \tan k\right) \cdot {\left(\frac{k}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t\_m}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e-92)
    (/ 2.0 (* (* t_m (tan k)) (pow (/ k (pow (cbrt l) 2.0)) 3.0)))
    (pow (* (/ l k) (pow t_m -1.5)) 2.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-92) {
		tmp = 2.0 / ((t_m * tan(k)) * pow((k / pow(cbrt(l), 2.0)), 3.0));
	} else {
		tmp = pow(((l / k) * pow(t_m, -1.5)), 2.0);
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 2.1e-92) {
		tmp = 2.0 / ((t_m * Math.tan(k)) * Math.pow((k / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t_m, -1.5)), 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e-92)
		tmp = Float64(2.0 / Float64(Float64(t_m * tan(k)) * (Float64(k / (cbrt(l) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(Float64(l / k) * (t_m ^ -1.5)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e-92], N[(2.0 / N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e-92

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in k around 0 55.8%

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

   6. Taylor expanded in k around inf 56.5%

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

      1. times-frac 57.9%

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

   8. Simplified 57.9%

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

      1. add-cube-cbrt 57.9%

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

      2. pow3 57.9%

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

      3. *-commutative 57.9%

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

      4. cbrt-prod 57.9%

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

      5. associate-/l* 57.9%

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

      6. tan-quot 57.9%

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

      7. cbrt-div 57.9%

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

      8. unpow3 57.9%

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

      9. add-cbrt-cube 64.9%

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

      10. unpow2 64.9%

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

      11. cbrt-prod 62.5%

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

      12. unpow2 62.5%

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

   10. Applied egg-rr 62.5%

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

       1. cube-prod 62.5%

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

       2. rem-cube-cbrt 62.5%

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

   12. Simplified 62.5%

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

   if 2.1e-92 < t

   1. Initial program 67.6%

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

   3. Simplified 67.5%

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

      1. add-sqr-sqrt 63.7%

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

   6. Applied egg-rr 69.0%

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

      1. unpow2 69.0%

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

      2. associate-/l* 68.9%

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

      3. associate-*l* 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in k around 0 67.0%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 67.0%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 67.0%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 67.0%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. pow1 67.0%

          \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-*r/ 67.0%

          \[\leadsto {\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       3. pow1/2 67.0%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       4. pow1/2 67.0%

          \[\leadsto {\left({\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       5. pow-prod-down 67.1%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       6. metadata-eval 67.1%

          \[\leadsto {\left({\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   13. Applied egg-rr 67.1%

       \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]
   14. Step-by-step derivation

       1. unpow1 67.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. pow-base-1 67.1%

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

       3. associate-*r/ 67.1%

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

       4. *-rgt-identity 67.1%

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

   15. Simplified 67.1%

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

       1. *-un-lft-identity 67.1%

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

       2. pow1/2 67.1%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)\right)}^{2} \]

       3. pow-flip 67.1%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)\right)}^{2} \]

       4. pow-pow 72.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)\right)}^{2} \]

       5. metadata-eval 72.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)\right)}^{2} \]

       6. metadata-eval 72.0%

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

   17. Applied egg-rr 72.0%

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

       1. *-lft-identity 72.0%

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

   19. Simplified 72.0%

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

Alternative 8: 69.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-16}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t\_m}^{-1.5}\right)}^{2}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \tan k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \tan k\right) \cdot \left({k}^{3} \cdot {\ell}^{-2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-16)
    (pow (* (/ l k) (pow t_m -1.5)) 2.0)
    (if (<= k 6e+109)
      (/ 2.0 (* (* 2.0 (tan k)) (* (sin k) (/ (pow t_m 3.0) (* l l)))))
      (/ 2.0 (* (* t_m (tan k)) (* (pow k 3.0) (pow l -2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-16) {
		tmp = pow(((l / k) * pow(t_m, -1.5)), 2.0);
	} else if (k <= 6e+109) {
		tmp = 2.0 / ((2.0 * tan(k)) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	} else {
		tmp = 2.0 / ((t_m * tan(k)) * (pow(k, 3.0) * pow(l, -2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6d-16) then
        tmp = ((l / k) * (t_m ** (-1.5d0))) ** 2.0d0
    else if (k <= 6d+109) then
        tmp = 2.0d0 / ((2.0d0 * tan(k)) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    else
        tmp = 2.0d0 / ((t_m * tan(k)) * ((k ** 3.0d0) * (l ** (-2.0d0))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 6e-16) {
		tmp = Math.pow(((l / k) * Math.pow(t_m, -1.5)), 2.0);
	} else if (k <= 6e+109) {
		tmp = 2.0 / ((2.0 * Math.tan(k)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	} else {
		tmp = 2.0 / ((t_m * Math.tan(k)) * (Math.pow(k, 3.0) * Math.pow(l, -2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6e-16:
		tmp = math.pow(((l / k) * math.pow(t_m, -1.5)), 2.0)
	elif k <= 6e+109:
		tmp = 2.0 / ((2.0 * math.tan(k)) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	else:
		tmp = 2.0 / ((t_m * math.tan(k)) * (math.pow(k, 3.0) * math.pow(l, -2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-16)
		tmp = Float64(Float64(l / k) * (t_m ^ -1.5)) ^ 2.0;
	elseif (k <= 6e+109)
		tmp = Float64(2.0 / Float64(Float64(2.0 * tan(k)) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * tan(k)) * Float64((k ^ 3.0) * (l ^ -2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6e-16)
		tmp = ((l / k) * (t_m ^ -1.5)) ^ 2.0;
	elseif (k <= 6e+109)
		tmp = 2.0 / ((2.0 * tan(k)) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	else
		tmp = 2.0 / ((t_m * tan(k)) * ((k ^ 3.0) * (l ^ -2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 6e-16], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 6e+109], N[(2.0 / N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.99999999999999987e-16

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-sqr-sqrt 46.6%

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

   6. Applied egg-rr 45.9%

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

      1. unpow2 45.9%

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

      2. associate-/l* 45.9%

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

      3. associate-*l* 41.7%

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

   8. Simplified 41.7%

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

   9. Taylor expanded in k around 0 42.4%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 42.4%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 42.4%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 42.4%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. pow1 42.4%

          \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-*r/ 42.4%

          \[\leadsto {\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       3. pow1/2 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       4. pow1/2 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       5. pow-prod-down 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       6. metadata-eval 42.4%

          \[\leadsto {\left({\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   13. Applied egg-rr 42.4%

       \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]
   14. Step-by-step derivation

       1. unpow1 42.4%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. pow-base-1 42.4%

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

       3. associate-*r/ 42.4%

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

       4. *-rgt-identity 42.4%

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

   15. Simplified 42.4%

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

       1. *-un-lft-identity 42.4%

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

       2. pow1/2 44.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)\right)}^{2} \]

       3. pow-flip 44.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)\right)}^{2} \]

       4. pow-pow 31.7%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)\right)}^{2} \]

       5. metadata-eval 31.7%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)\right)}^{2} \]

       6. metadata-eval 31.7%

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

   17. Applied egg-rr 31.7%

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

       1. *-lft-identity 31.7%

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

   19. Simplified 31.7%

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

   if 5.99999999999999987e-16 < k < 6.00000000000000031e109

   1. Initial program 47.8%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in k around 0 67.7%

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

   if 6.00000000000000031e109 < k

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in k around 0 35.8%

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

   6. Taylor expanded in k around inf 49.5%

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

      1. times-frac 49.5%

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

   8. Simplified 49.5%

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

      1. div-inv 49.5%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

      2. div-inv 49.5%

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

      3. pow-flip 49.6%

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

      4. metadata-eval 49.6%

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

      5. associate-/l* 49.6%

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

      6. tan-quot 49.6%

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

   10. Applied egg-rr 49.6%

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

       1. associate-*r/ 49.6%

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

       2. metadata-eval 49.6%

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

   12. Simplified 49.6%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-16}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \tan k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \tan k\right) \cdot \left({k}^{3} \cdot {\ell}^{-2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \tan k\right) \cdot \left({k}^{3} \cdot {\ell}^{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t\_m}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.65e-95)
    (/ 2.0 (* (* t_m (tan k)) (* (pow k 3.0) (pow l -2.0))))
    (pow (* (/ l k) (pow t_m -1.5)) 2.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e-95) {
		tmp = 2.0 / ((t_m * tan(k)) * (pow(k, 3.0) * pow(l, -2.0)));
	} else {
		tmp = pow(((l / k) * pow(t_m, -1.5)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.65d-95) then
        tmp = 2.0d0 / ((t_m * tan(k)) * ((k ** 3.0d0) * (l ** (-2.0d0))))
    else
        tmp = ((l / k) * (t_m ** (-1.5d0))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 1.65e-95) {
		tmp = 2.0 / ((t_m * Math.tan(k)) * (Math.pow(k, 3.0) * Math.pow(l, -2.0)));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t_m, -1.5)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.65e-95:
		tmp = 2.0 / ((t_m * math.tan(k)) * (math.pow(k, 3.0) * math.pow(l, -2.0)))
	else:
		tmp = math.pow(((l / k) * math.pow(t_m, -1.5)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.65e-95)
		tmp = Float64(2.0 / Float64(Float64(t_m * tan(k)) * Float64((k ^ 3.0) * (l ^ -2.0))));
	else
		tmp = Float64(Float64(l / k) * (t_m ^ -1.5)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.65e-95)
		tmp = 2.0 / ((t_m * tan(k)) * ((k ^ 3.0) * (l ^ -2.0)));
	else
		tmp = ((l / k) * (t_m ^ -1.5)) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.65e-95], N[(2.0 / N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.65e-95

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in k around 0 55.8%

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

   6. Taylor expanded in k around inf 56.5%

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

      1. times-frac 57.9%

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

   8. Simplified 57.9%

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

      1. div-inv 57.9%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

      2. div-inv 57.8%

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

      3. pow-flip 57.9%

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

      4. metadata-eval 57.9%

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

      5. associate-/l* 57.9%

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

      6. tan-quot 57.9%

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

   10. Applied egg-rr 57.9%

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

       1. associate-*r/ 57.9%

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

       2. metadata-eval 57.9%

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

   12. Simplified 57.9%

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

   if 1.65e-95 < t

   1. Initial program 67.6%

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

   3. Simplified 67.5%

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

      1. add-sqr-sqrt 63.7%

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

   6. Applied egg-rr 69.0%

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

      1. unpow2 69.0%

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

      2. associate-/l* 68.9%

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

      3. associate-*l* 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in k around 0 67.0%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 67.0%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 67.0%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 67.0%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. pow1 67.0%

          \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-*r/ 67.0%

          \[\leadsto {\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       3. pow1/2 67.0%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       4. pow1/2 67.0%

          \[\leadsto {\left({\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       5. pow-prod-down 67.1%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       6. metadata-eval 67.1%

          \[\leadsto {\left({\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   13. Applied egg-rr 67.1%

       \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]
   14. Step-by-step derivation

       1. unpow1 67.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. pow-base-1 67.1%

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

       3. associate-*r/ 67.1%

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

       4. *-rgt-identity 67.1%

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

   15. Simplified 67.1%

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

       1. *-un-lft-identity 67.1%

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

       2. pow1/2 67.1%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)\right)}^{2} \]

       3. pow-flip 67.1%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)\right)}^{2} \]

       4. pow-pow 72.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)\right)}^{2} \]

       5. metadata-eval 72.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)\right)}^{2} \]

       6. metadata-eval 72.0%

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

   17. Applied egg-rr 72.0%

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

       1. *-lft-identity 72.0%

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

   19. Simplified 72.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(t \cdot \tan k\right) \cdot \left({k}^{3} \cdot {\ell}^{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t\_m}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-89)
    (/ 2.0 (* (/ (pow k 3.0) (pow l 2.0)) (* t_m k)))
    (pow (* (/ l k) (pow t_m -1.5)) 2.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-89) {
		tmp = 2.0 / ((pow(k, 3.0) / pow(l, 2.0)) * (t_m * k));
	} else {
		tmp = pow(((l / k) * pow(t_m, -1.5)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.1d-89) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l ** 2.0d0)) * (t_m * k))
    else
        tmp = ((l / k) * (t_m ** (-1.5d0))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 3.1e-89) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / Math.pow(l, 2.0)) * (t_m * k));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t_m, -1.5)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.1e-89:
		tmp = 2.0 / ((math.pow(k, 3.0) / math.pow(l, 2.0)) * (t_m * k))
	else:
		tmp = math.pow(((l / k) * math.pow(t_m, -1.5)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.1e-89)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / (l ^ 2.0)) * Float64(t_m * k)));
	else
		tmp = Float64(Float64(l / k) * (t_m ^ -1.5)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.1e-89)
		tmp = 2.0 / (((k ^ 3.0) / (l ^ 2.0)) * (t_m * k));
	else
		tmp = ((l / k) * (t_m ^ -1.5)) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-89], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.09999999999999996e-89

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in k around 0 55.8%

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

   6. Taylor expanded in k around inf 56.5%

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

      1. times-frac 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in k around 0 56.1%

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

   if 3.09999999999999996e-89 < t

   1. Initial program 67.6%

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

   3. Simplified 67.5%

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

      1. add-sqr-sqrt 63.7%

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

   6. Applied egg-rr 69.0%

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

      1. unpow2 69.0%

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

      2. associate-/l* 68.9%

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

      3. associate-*l* 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in k around 0 67.0%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 67.0%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 67.0%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 67.0%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. pow1 67.0%

          \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-*r/ 67.0%

          \[\leadsto {\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       3. pow1/2 67.0%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       4. pow1/2 67.0%

          \[\leadsto {\left({\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       5. pow-prod-down 67.1%

          \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       6. metadata-eval 67.1%

          \[\leadsto {\left({\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   13. Applied egg-rr 67.1%

       \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]
   14. Step-by-step derivation

       1. unpow1 67.1%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. pow-base-1 67.1%

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

       3. associate-*r/ 67.1%

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

       4. *-rgt-identity 67.1%

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

   15. Simplified 67.1%

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

       1. *-un-lft-identity 67.1%

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

       2. pow1/2 67.1%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)\right)}^{2} \]

       3. pow-flip 67.1%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)\right)}^{2} \]

       4. pow-pow 72.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)\right)}^{2} \]

       5. metadata-eval 72.0%

          \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)\right)}^{2} \]

       6. metadata-eval 72.0%

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

   17. Applied egg-rr 72.0%

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

       1. *-lft-identity 72.0%

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

   19. Simplified 72.0%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.65 \cdot 10^{-118}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{1}{\frac{\ell}{{t\_m}^{2}}} \cdot \frac{t\_m}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3.65e-118)
    (/ (pow (/ l k) 2.0) (pow t_m 3.0))
    (/ 2.0 (* (* (/ 1.0 (/ l (pow t_m 2.0))) (/ t_m l)) (* 2.0 (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.65e-118) {
		tmp = pow((l / k), 2.0) / pow(t_m, 3.0);
	} else {
		tmp = 2.0 / (((1.0 / (l / pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.65d-118) then
        tmp = ((l / k) ** 2.0d0) / (t_m ** 3.0d0)
    else
        tmp = 2.0d0 / (((1.0d0 / (l / (t_m ** 2.0d0))) * (t_m / l)) * (2.0d0 * (k * k)))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (l <= 3.65e-118) {
		tmp = Math.pow((l / k), 2.0) / Math.pow(t_m, 3.0);
	} else {
		tmp = 2.0 / (((1.0 / (l / Math.pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.65e-118:
		tmp = math.pow((l / k), 2.0) / math.pow(t_m, 3.0)
	else:
		tmp = 2.0 / (((1.0 / (l / math.pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.65e-118)
		tmp = Float64((Float64(l / k) ^ 2.0) / (t_m ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(1.0 / Float64(l / (t_m ^ 2.0))) * Float64(t_m / l)) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.65e-118)
		tmp = ((l / k) ^ 2.0) / (t_m ^ 3.0);
	else
		tmp = 2.0 / (((1.0 / (l / (t_m ^ 2.0))) * (t_m / l)) * (2.0 * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[l, 3.65e-118], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.65e-118

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

      1. add-cube-cbrt 58.1%

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

      2. pow3 58.1%

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

      3. associate-/r* 61.9%

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

      4. *-commutative 61.9%

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

      5. cbrt-prod 61.9%

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

      6. associate-/r* 58.1%

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

      7. cbrt-div 59.1%

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

      8. rem-cbrt-cube 66.1%

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

      9. cbrt-prod 75.9%

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

      10. pow2 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. pow1 75.9%

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

      2. associate-+r+ 75.9%

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

      3. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. unpow1 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in k around 0 52.5%

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

       1. associate-/r* 53.5%

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

       2. unpow2 53.5%

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

       3. unpow2 53.5%

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

       4. times-frac 66.2%

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

       5. *-rgt-identity 66.2%

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{{t}^{3}} \]

       6. associate-*r/ 66.2%

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

       7. pow-base-1 66.2%

          \[\leadsto \frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\color{blue}{{1}^{0.5}}}{k}\right)}{{t}^{3}} \]

       8. *-rgt-identity 66.2%

          \[\leadsto \frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}{{t}^{3}} \]

       9. associate-*r/ 66.2%

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}{{t}^{3}} \]

       10. pow-base-1 66.2%

           \[\leadsto \frac{\left(\ell \cdot \frac{\color{blue}{{1}^{0.5}}}{k}\right) \cdot \left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}{{t}^{3}} \]

       11. unpow2 66.2%

           \[\leadsto \frac{\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{2}}}{{t}^{3}} \]

       12. pow-base-1 66.2%

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

       13. associate-*r/ 66.2%

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

       14. *-rgt-identity 66.2%

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

   13. Simplified 66.2%

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

   if 3.65e-118 < l

   1. Initial program 64.2%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in k around 0 60.1%

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

      1. unpow2 60.1%

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

   7. Applied egg-rr 60.1%

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

      1. associate-/r* 58.4%

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

      2. unpow3 58.4%

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

      3. times-frac 62.5%

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

      4. pow2 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. clear-num 62.5%

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

       2. inv-pow 62.5%

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

   11. Applied egg-rr 62.5%

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

       1. unpow-1 62.5%

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

   13. Simplified 62.5%

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

Alternative 12: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot {t\_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{1}{\frac{\ell}{{t\_m}^{2}}} \cdot \frac{t\_m}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3.6e-118)
    (* (pow (/ l k) 2.0) (pow t_m -3.0))
    (/ 2.0 (* (* (/ 1.0 (/ l (pow t_m 2.0))) (/ t_m l)) (* 2.0 (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.6e-118) {
		tmp = pow((l / k), 2.0) * pow(t_m, -3.0);
	} else {
		tmp = 2.0 / (((1.0 / (l / pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.6d-118) then
        tmp = ((l / k) ** 2.0d0) * (t_m ** (-3.0d0))
    else
        tmp = 2.0d0 / (((1.0d0 / (l / (t_m ** 2.0d0))) * (t_m / l)) * (2.0d0 * (k * k)))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (l <= 3.6e-118) {
		tmp = Math.pow((l / k), 2.0) * Math.pow(t_m, -3.0);
	} else {
		tmp = 2.0 / (((1.0 / (l / Math.pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.6e-118:
		tmp = math.pow((l / k), 2.0) * math.pow(t_m, -3.0)
	else:
		tmp = 2.0 / (((1.0 / (l / math.pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.6e-118)
		tmp = Float64((Float64(l / k) ^ 2.0) * (t_m ^ -3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(1.0 / Float64(l / (t_m ^ 2.0))) * Float64(t_m / l)) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.6e-118)
		tmp = ((l / k) ^ 2.0) * (t_m ^ -3.0);
	else
		tmp = 2.0 / (((1.0 / (l / (t_m ^ 2.0))) * (t_m / l)) * (2.0 * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[l, 3.6e-118], N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.6000000000000002e-118

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

      1. add-sqr-sqrt 47.0%

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

   6. Applied egg-rr 46.0%

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

      1. unpow2 46.0%

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

      2. associate-/l* 46.0%

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

      3. associate-*l* 41.8%

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

   8. Simplified 41.8%

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

   9. Taylor expanded in k around 0 41.5%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 41.5%

          \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       2. associate-/l* 41.5%

          \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

   11. Simplified 41.5%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. associate-*l* 41.1%

          \[\leadsto {\color{blue}{\left(\ell \cdot \left(\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)\right)}}^{2} \]

       2. unpow-prod-down 39.0%

          \[\leadsto \color{blue}{{\ell}^{2} \cdot {\left(\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2}} \]

       3. associate-*r/ 39.0%

          \[\leadsto {\ell}^{2} \cdot {\left(\color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       4. pow1/2 39.0%

          \[\leadsto {\ell}^{2} \cdot {\left(\frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       5. pow1/2 39.0%

          \[\leadsto {\ell}^{2} \cdot {\left(\frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       6. pow-prod-down 39.0%

          \[\leadsto {\ell}^{2} \cdot {\left(\frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       7. metadata-eval 39.0%

          \[\leadsto {\ell}^{2} \cdot {\left(\frac{{\color{blue}{1}}^{0.5}}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

       8. pow1/2 41.5%

          \[\leadsto {\ell}^{2} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)}^{2} \]

       9. pow-flip 41.5%

          \[\leadsto {\ell}^{2} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)}^{2} \]

       10. pow-pow 27.4%

           \[\leadsto {\ell}^{2} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)}^{2} \]

       11. metadata-eval 27.4%

           \[\leadsto {\ell}^{2} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)}^{2} \]

       12. metadata-eval 27.4%

           \[\leadsto {\ell}^{2} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot {t}^{\color{blue}{-1.5}}\right)}^{2} \]

   13. Applied egg-rr 27.4%

       \[\leadsto \color{blue}{{\ell}^{2} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)}^{2}} \]
   14. Step-by-step derivation

       1. unpow2 27.4%

          \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot {\left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)}^{2} \]

       2. unpow2 27.4%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right) \cdot \left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)\right)} \]

       3. swap-sqr 29.4%

          \[\leadsto \color{blue}{\left(\ell \cdot \left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)\right) \cdot \left(\ell \cdot \left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)\right)} \]

       4. associate-*l* 29.4%

          \[\leadsto \color{blue}{\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{-1.5}\right)} \cdot \left(\ell \cdot \left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)\right) \]

       5. associate-*l* 30.0%

          \[\leadsto \left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{-1.5}\right) \cdot \color{blue}{\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{-1.5}\right)} \]

       6. swap-sqr 28.7%

          \[\leadsto \color{blue}{\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot \left(\ell \cdot \frac{{1}^{0.5}}{k}\right)\right) \cdot \left({t}^{-1.5} \cdot {t}^{-1.5}\right)} \]

       7. unpow2 28.7%

          \[\leadsto \color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{2}} \cdot \left({t}^{-1.5} \cdot {t}^{-1.5}\right) \]

       8. pow-sqr 65.6%

          \[\leadsto {\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{2} \cdot \color{blue}{{t}^{\left(2 \cdot -1.5\right)}} \]

       9. metadata-eval 65.6%

          \[\leadsto {\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{2} \cdot {t}^{\color{blue}{-3}} \]

       10. *-commutative 65.6%

           \[\leadsto \color{blue}{{t}^{-3} \cdot {\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{2}} \]

       11. pow-base-1 65.6%

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

       12. associate-*r/ 65.6%

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

       13. *-rgt-identity 65.6%

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

   15. Simplified 65.6%

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

   if 3.6000000000000002e-118 < l

   1. Initial program 64.2%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in k around 0 60.1%

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

      1. unpow2 60.1%

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

   7. Applied egg-rr 60.1%

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

      1. associate-/r* 58.4%

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

      2. unpow3 58.4%

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

      3. times-frac 62.5%

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

      4. pow2 62.5%

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

   9. Applied egg-rr 62.5%

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

       1. clear-num 62.5%

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

       2. inv-pow 62.5%

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

   11. Applied egg-rr 62.5%

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

       1. unpow-1 62.5%

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

   13. Simplified 62.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{1}{\frac{\ell}{{t}^{2}}} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\frac{\ell}{k} \cdot {t\_m}^{-1.5}\right)}^{2} \end{array} \]

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * pow(((l / k) * pow(t_m, -1.5)), 2.0);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / k) * (t_m ** (-1.5d0))) ** 2.0d0)
end function

Java

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) {
	return t_s * Math.pow(((l / k) * Math.pow(t_m, -1.5)), 2.0);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * math.pow(((l / k) * math.pow(t_m, -1.5)), 2.0)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(Float64(l / k) * (t_m ^ -1.5)) ^ 2.0))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / k) * (t_m ^ -1.5)) ^ 2.0);
end

Wolfram

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_] := N[(t$95$s * N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

3. Simplified 60.1%

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

   1. add-sqr-sqrt 44.8%

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

6. Applied egg-rr 46.5%

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

   1. unpow2 46.5%

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

   2. associate-/l* 46.5%

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

   3. associate-*l* 43.3%

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

8. Simplified 43.3%

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

9. Taylor expanded in k around 0 40.1%

   \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
10. Step-by-step derivation

    1. associate-/l* 40.1%

       \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    2. associate-/l* 40.1%

       \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

11. Simplified 40.1%

    \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
12. Step-by-step derivation

    1. pow1 40.1%

       \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    2. associate-*r/ 40.1%

       \[\leadsto {\left({\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    3. pow1/2 40.1%

       \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    4. pow1/2 40.1%

       \[\leadsto {\left({\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    5. pow-prod-down 40.1%

       \[\leadsto {\left({\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    6. metadata-eval 40.1%

       \[\leadsto {\left({\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right)}^{1} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

13. Applied egg-rr 40.1%

    \[\leadsto {\left(\color{blue}{{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)}^{1}} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]
14. Step-by-step derivation

    1. unpow1 40.1%

       \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{{1}^{0.5}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    2. pow-base-1 40.1%

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

    3. associate-*r/ 40.1%

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

    4. *-rgt-identity 40.1%

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

15. Simplified 40.1%

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

    1. *-un-lft-identity 40.1%

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

    2. pow1/2 42.5%

       \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)\right)}^{2} \]

    3. pow-flip 42.5%

       \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)\right)}^{2} \]

    4. pow-pow 29.4%

       \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)\right)}^{2} \]

    5. metadata-eval 29.4%

       \[\leadsto {\left(\frac{\ell}{k} \cdot \left(1 \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)\right)}^{2} \]

    6. metadata-eval 29.4%

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

17. Applied egg-rr 29.4%

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

    1. *-lft-identity 29.4%

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

19. Simplified 29.4%

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

Alternative 14: 67.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\ell \cdot \frac{{t\_m}^{-1.5}}{k}\right)}^{2} \end{array} \]

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * pow((l * (pow(t_m, -1.5) / k)), 2.0);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * ((t_m ** (-1.5d0)) / k)) ** 2.0d0)
end function

Java

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) {
	return t_s * Math.pow((l * (Math.pow(t_m, -1.5) / k)), 2.0);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * math.pow((l * (math.pow(t_m, -1.5) / k)), 2.0)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(l * Float64((t_m ^ -1.5) / k)) ^ 2.0))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * ((t_m ^ -1.5) / k)) ^ 2.0);
end

Wolfram

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_] := N[(t$95$s * N[Power[N[(l * N[(N[Power[t$95$m, -1.5], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

3. Simplified 60.1%

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

   1. add-sqr-sqrt 44.8%

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

6. Applied egg-rr 46.5%

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

   1. unpow2 46.5%

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

   2. associate-/l* 46.5%

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

   3. associate-*l* 43.3%

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

8. Simplified 43.3%

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

9. Taylor expanded in k around 0 40.1%

   \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
10. Step-by-step derivation

    1. associate-/l* 40.1%

       \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{k}\right)} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

    2. associate-/l* 40.1%

       \[\leadsto {\left(\left(\ell \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2} \]

11. Simplified 40.1%

    \[\leadsto {\color{blue}{\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]
12. Step-by-step derivation

    1. pow1 40.1%

       \[\leadsto {\color{blue}{\left({\left(\left(\ell \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{k}\right)\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{1}\right)}}^{2} \]

    2. associate-*r/ 40.1%

       \[\leadsto {\left({\left(\left(\ell \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{k}}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{1}\right)}^{2} \]

    3. pow1/2 40.1%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{\color{blue}{{0.5}^{0.5}} \cdot \sqrt{2}}{k}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{1}\right)}^{2} \]

    4. pow1/2 40.1%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{{0.5}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{k}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{1}\right)}^{2} \]

    5. pow-prod-down 40.1%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{\color{blue}{{\left(0.5 \cdot 2\right)}^{0.5}}}{k}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{1}\right)}^{2} \]

    6. metadata-eval 40.1%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{{\color{blue}{1}}^{0.5}}{k}\right) \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{1}\right)}^{2} \]

    7. pow1/2 42.5%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot \color{blue}{{\left(\frac{1}{{t}^{3}}\right)}^{0.5}}\right)}^{1}\right)}^{2} \]

    8. pow-flip 42.5%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {\color{blue}{\left({t}^{\left(-3\right)}\right)}}^{0.5}\right)}^{1}\right)}^{2} \]

    9. pow-pow 29.4%

       \[\leadsto {\left({\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot \color{blue}{{t}^{\left(\left(-3\right) \cdot 0.5\right)}}\right)}^{1}\right)}^{2} \]

    10. metadata-eval 29.4%

        \[\leadsto {\left({\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{\left(\color{blue}{-3} \cdot 0.5\right)}\right)}^{1}\right)}^{2} \]

    11. metadata-eval 29.4%

        \[\leadsto {\left({\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{\color{blue}{-1.5}}\right)}^{1}\right)}^{2} \]

13. Applied egg-rr 29.4%

    \[\leadsto {\color{blue}{\left({\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{-1.5}\right)}^{1}\right)}}^{2} \]
14. Step-by-step derivation

    1. unpow1 29.4%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{{1}^{0.5}}{k}\right) \cdot {t}^{-1.5}\right)}}^{2} \]

    2. associate-*l* 29.1%

       \[\leadsto {\color{blue}{\left(\ell \cdot \left(\frac{{1}^{0.5}}{k} \cdot {t}^{-1.5}\right)\right)}}^{2} \]

    3. pow-base-1 29.1%

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

    4. associate-*l/ 29.1%

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

    5. *-lft-identity 29.1%

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

15. Simplified 29.1%

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

Alternative 15: 57.9% accurate, 3.6× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (/ 1.0 (/ l (pow t_m 2.0))) (/ t_m l)) (* 2.0 (* k k))))))

C

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

Fortran

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

Java

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) {
	return t_s * (2.0 / (((1.0 / (l / Math.pow(t_m, 2.0))) * (t_m / l)) * (2.0 * (k * k))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_] := N[(t$95$s * N[(2.0 / N[(N[(N[(1.0 / N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

3. Simplified 58.5%

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

5. Taylor expanded in k around 0 56.8%

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

   1. unpow2 56.8%

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

7. Applied egg-rr 56.8%

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

   1. associate-/r* 54.4%

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

   2. unpow3 54.4%

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

   3. times-frac 58.7%

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

   4. pow2 58.7%

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

9. Applied egg-rr 58.7%

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

    1. clear-num 58.7%

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

    2. inv-pow 58.7%

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

11. Applied egg-rr 58.7%

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

    1. unpow-1 58.7%

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

13. Simplified 58.7%

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

Alternative 16: 57.9% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

3. Simplified 58.5%

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

5. Taylor expanded in k around 0 56.8%

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

   1. unpow2 56.8%

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

7. Applied egg-rr 56.8%

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

   1. associate-/r* 54.4%

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

   2. unpow3 54.4%

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

   3. times-frac 58.7%

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

   4. pow2 58.7%

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

9. Applied egg-rr 58.7%

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

    1. unpow2 58.7%

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

11. Applied egg-rr 58.7%

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

12. Final simplification 58.7%

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

Reproduce

herbie shell --seed 2024181 
(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))))