Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.8% → 88.8%

Time: 23.6s

Alternatives: 21

Speedup: 3.8×

• 27.3% 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: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.7e-36)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (/
     2.0
     (*
      (pow (/ (* t_m (cbrt (sin k))) (pow (cbrt l) 2.0)) 3.0)
      (* (tan k) (+ 1.0 (+ 1.0 (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 <= 1.7e-36) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else {
		tmp = 2.0 / (pow(((t_m * cbrt(sin(k))) / pow(cbrt(l), 2.0)), 3.0) * (tan(k) * (1.0 + (1.0 + 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 <= 1.7e-36) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (Math.tan(k) * (1.0 + (1.0 + 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 <= 1.7e-36)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * cbrt(sin(k))) / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + (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, 1.7e-36], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7000000000000001e-36

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 1.7000000000000001e-36 < t

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

      1. add-cube-cbrt 67.5%

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

      2. pow3 67.5%

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

      3. *-commutative 67.5%

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

      4. cbrt-prod 67.5%

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

      5. cbrt-div 68.7%

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

      6. rem-cbrt-cube 73.1%

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

      7. cbrt-prod 92.6%

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

      8. pow2 92.6%

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

   6. Applied egg-rr 92.6%

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

      1. *-commutative 92.6%

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

   8. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   10. Applied egg-rr 92.7%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

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 1.8e-34)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 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 <= 1.8e-34) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 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 <= 1.8e-34) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 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 <= 1.8e-34)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 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, 1.8e-34], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000004e-34

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 1.80000000000000004e-34 < t

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

      1. add-cube-cbrt 67.5%

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

      2. pow3 67.5%

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

      3. *-commutative 67.5%

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

      4. cbrt-prod 67.5%

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

      5. cbrt-div 68.7%

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

      6. rem-cbrt-cube 73.1%

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

      7. cbrt-prod 92.6%

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

      8. pow2 92.6%

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

   6. Applied egg-rr 92.6%

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

      1. *-commutative 92.6%

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

   8. Simplified 92.6%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

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 3.3e-35)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
      (* (sin k) (pow (/ t_m (pow (cbrt l) 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 <= 3.3e-35) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * (sin(k) * pow((t_m / pow(cbrt(l), 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 <= 3.3e-35) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 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 <= 3.3e-35)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 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, 3.3e-35], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.3e-35

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 3.3e-35 < t

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

      1. add-cube-cbrt 67.5%

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

      2. pow3 67.5%

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

      3. cbrt-div 67.6%

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

      4. rem-cbrt-cube 72.0%

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

      5. cbrt-prod 87.8%

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

      6. pow2 87.8%

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

   6. Applied egg-rr 87.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 7e-36)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (if (<= t_m 1.7e+130)
      (/
       2.0
       (/
        (*
         (* t_m (/ (pow t_m 2.0) l))
         (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (tan k))))
        l))
      (/
       2.0
       (*
        (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
        (* 2.0 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 (t_m <= 7e-36) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else if (t_m <= 1.7e+130) {
		tmp = 2.0 / (((t_m * (pow(t_m, 2.0) / l)) * ((2.0 + pow((k / t_m), 2.0)) * (sin(k) * tan(k)))) / l);
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	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 <= 7e-36) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else if (t_m <= 1.7e+130) {
		tmp = 2.0 / (((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * Math.tan(k)))) / l);
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * 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 (t_m <= 7e-36)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	elseif (t_m <= 1.7e+130)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * tan(k)))) / l));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	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, 7e-36], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+130], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.9999999999999999e-36

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 6.9999999999999999e-36 < t < 1.7e130

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

      1. associate-*l* 71.7%

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

      2. associate-/r* 74.1%

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

      3. associate-+r+ 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. associate-*l/ 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. cube-mult 79.2%

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

      2. *-un-lft-identity 79.2%

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

      3. times-frac 89.2%

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

      4. pow2 89.2%

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

   8. Applied egg-rr 89.2%

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

   if 1.7e130 < t

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

      3. *-commutative 61.2%

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

      4. cbrt-prod 61.2%

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

      5. cbrt-div 61.2%

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

      6. rem-cbrt-cube 67.1%

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

      7. cbrt-prod 91.9%

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

      8. pow2 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. *-commutative 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in k around 0 82.4%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 1.55 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 2.3e-36)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (if (<= t_m 1.55e+132)
      (/
       2.0
       (/
        (*
         (* t_m (/ (pow t_m 2.0) l))
         (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (tan k))))
        l))
      (/
       2.0
       (*
        (pow (/ (* t_m (cbrt (sin k))) (pow (cbrt l) 2.0)) 3.0)
        (* 2.0 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 (t_m <= 2.3e-36) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else if (t_m <= 1.55e+132) {
		tmp = 2.0 / (((t_m * (pow(t_m, 2.0) / l)) * ((2.0 + pow((k / t_m), 2.0)) * (sin(k) * tan(k)))) / l);
	} else {
		tmp = 2.0 / (pow(((t_m * cbrt(sin(k))) / pow(cbrt(l), 2.0)), 3.0) * (2.0 * k));
	}
	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 <= 2.3e-36) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else if (t_m <= 1.55e+132) {
		tmp = 2.0 / (((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * Math.tan(k)))) / l);
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * 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 (t_m <= 2.3e-36)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	elseif (t_m <= 1.55e+132)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * tan(k)))) / l));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * cbrt(sin(k))) / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * k)));
	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, 2.3e-36], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.55e+132], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.29999999999999996e-36

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 2.29999999999999996e-36 < t < 1.5499999999999999e132

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

      1. associate-*l* 71.7%

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

      2. associate-/r* 74.1%

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

      3. associate-+r+ 74.1%

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

      4. metadata-eval 74.1%

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

      5. associate-*l* 74.1%

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

      6. associate-*l/ 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. cube-mult 79.2%

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

      2. *-un-lft-identity 79.2%

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

      3. times-frac 89.2%

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

      4. pow2 89.2%

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

   8. Applied egg-rr 89.2%

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

   if 1.5499999999999999e132 < t

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

      3. *-commutative 61.2%

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

      4. cbrt-prod 61.2%

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

      5. cbrt-div 61.2%

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

      6. rem-cbrt-cube 67.1%

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

      7. cbrt-prod 91.9%

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

      8. pow2 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. *-commutative 91.9%

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

   8. Simplified 91.9%

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

      1. associate-*l/ 92.0%

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

   10. Applied egg-rr 92.0%

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

   11. Taylor expanded in k around 0 82.5%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.45e-34)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (* (sin k) (pow (/ (pow t_m 1.5) 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 (t_m <= 1.45e-34) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * pow((pow(t_m, 1.5) / 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 (t_m <= 1.45d-34) then
        tmp = 2.0d0 / ((1.0d0 / l) * (((sqrt(t_m) * (k * sin(k))) ** 2.0d0) / (l * cos(k))))
    else
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * (sin(k) * (((t_m ** 1.5d0) / 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 (t_m <= 1.45e-34) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / 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 t_m <= 1.45e-34:
		tmp = 2.0 / ((1.0 / l) * (math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0) / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / 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 (t_m <= 1.45e-34)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / 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 (t_m <= 1.45e-34)
		tmp = 2.0 / ((1.0 / l) * (((sqrt(t_m) * (k * sin(k))) ^ 2.0) / (l * cos(k))));
	else
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * (sin(k) * (((t_m ^ 1.5) / 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[t$95$m, 1.45e-34], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-34

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 1.4500000000000001e-34 < t

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

      1. add-sqr-sqrt 38.4%

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

      2. pow2 38.4%

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

      3. *-commutative 38.4%

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

      4. sqrt-prod 38.4%

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

      5. sqrt-div 39.7%

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

      6. sqrt-pow1 42.5%

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

      7. metadata-eval 42.5%

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

      8. sqrt-prod 27.4%

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

      9. add-sqr-sqrt 51.1%

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

   6. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

   8. Simplified 51.1%

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

      1. pow1 51.1%

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

      2. *-commutative 51.1%

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

      3. associate-+r+ 51.1%

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

      4. metadata-eval 51.1%

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

      5. *-commutative 51.1%

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

      6. unpow-prod-down 48.7%

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

      7. pow2 48.7%

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

      8. add-sqr-sqrt 87.0%

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

   10. Applied egg-rr 87.0%

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

       1. unpow1 87.0%

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

   12. Simplified 87.0%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 5.5e-37)
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos k)))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
      (* (sin k) (* (/ (pow t_m 2.0) l) (/ 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) {
	double tmp;
	if (t_m <= 5.5e-37) {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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 <= 5.5d-37) then
        tmp = 2.0d0 / ((1.0d0 / l) * (((sqrt(t_m) * (k * sin(k))) ** 2.0d0) / (l * cos(k))))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0)))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    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 <= 5.5e-37) {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	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 <= 5.5e-37:
		tmp = 2.0 / ((1.0 / l) * (math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0) / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + math.pow((k / t_m), 2.0)))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	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 <= 5.5e-37)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	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 <= 5.5e-37)
		tmp = 2.0 / ((1.0 / l) * (((sqrt(t_m) * (k * sin(k))) ^ 2.0) / (l * cos(k))));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) ^ 2.0)))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	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, 5.5e-37], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.4999999999999998e-37

   1. Initial program 46.6%

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

   3. Simplified 46.6%

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

      1. associate-*l* 44.8%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.1%

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

      7. clear-num 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-/r/ 53.1%

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

      2. associate-*l* 53.1%

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

   8. Simplified 53.1%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. div-inv 76.8%

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

       2. add-sqr-sqrt 32.0%

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

       3. pow2 32.0%

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

       4. sqrt-prod 32.0%

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

       5. sqrt-pow1 34.6%

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

       6. metadata-eval 34.6%

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

       7. pow1 34.6%

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

       8. *-commutative 34.6%

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

       9. sqrt-prod 34.5%

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

       10. sqrt-pow1 34.5%

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

       11. metadata-eval 34.5%

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

       12. pow1 34.5%

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

   11. Applied egg-rr 34.5%

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

       1. associate-*r/ 34.5%

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

       2. *-commutative 34.5%

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

       3. *-rgt-identity 34.5%

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

       4. *-commutative 34.5%

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

       5. *-commutative 34.5%

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

       6. associate-*l* 34.5%

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

   13. Simplified 34.5%

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

   if 5.4999999999999998e-37 < t

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

      1. unpow3 67.7%

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

      2. times-frac 79.5%

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

      3. pow2 79.5%

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

   6. Applied egg-rr 79.5%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(k \cdot \left(\sqrt{t\_m} \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

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 1.8e-28)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* k (* (sqrt t_m) (sin k))) 2.0) (* l (cos 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 (k <= 1.8e-28) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((1.0 / l) * (pow((k * (sqrt(t_m) * sin(k))), 2.0) / (l * cos(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 (k <= 1.8d-28) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = 2.0d0 / ((1.0d0 / l) * (((k * (sqrt(t_m) * sin(k))) ** 2.0d0) / (l * cos(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 (k <= 1.8e-28) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((k * (Math.sqrt(t_m) * Math.sin(k))), 2.0) / (l * Math.cos(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 k <= 1.8e-28:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = 2.0 / ((1.0 / l) * (math.pow((k * (math.sqrt(t_m) * math.sin(k))), 2.0) / (l * math.cos(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 (k <= 1.8e-28)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(k * Float64(sqrt(t_m) * sin(k))) ^ 2.0) / Float64(l * cos(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 (k <= 1.8e-28)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = 2.0 / ((1.0 / l) * (((k * (sqrt(t_m) * sin(k))) ^ 2.0) / (l * cos(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[k, 1.8e-28], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7999999999999999e-28

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 30.2%

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

      2. pow2 30.2%

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

      3. *-commutative 30.2%

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

      4. sqrt-prod 16.6%

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

      5. sqrt-div 17.1%

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

      6. sqrt-pow1 19.6%

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

      7. metadata-eval 19.6%

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

      8. sqrt-prod 13.4%

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

      9. add-sqr-sqrt 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. *-commutative 26.8%

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

   8. Simplified 26.8%

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

   9. Taylor expanded in k around 0 23.6%

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

   if 1.7999999999999999e-28 < k

   1. Initial program 47.2%

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

   3. Simplified 47.2%

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

      1. associate-*l* 47.3%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.5%

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

      7. clear-num 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-/r/ 53.5%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 80.6%

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

       1. pow1 80.6%

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

       2. add-sqr-sqrt 38.2%

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

       3. pow2 38.2%

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

       4. sqrt-prod 38.1%

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

       5. sqrt-pow1 43.7%

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

       6. metadata-eval 43.7%

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

       7. pow1 43.7%

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

       8. *-commutative 43.7%

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

       9. sqrt-prod 43.7%

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

       10. sqrt-pow1 43.7%

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

       11. metadata-eval 43.7%

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

       12. pow1 43.7%

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

   11. Applied egg-rr 43.7%

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

       1. unpow1 43.7%

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

   13. Simplified 43.7%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(k \cdot \left(\sqrt{t} \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

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 5e-29)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (/
     2.0
     (* (/ 1.0 l) (/ (pow (* (sqrt t_m) (* k (sin k))) 2.0) (* l (cos 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 (k <= 5e-29) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((1.0 / l) * (pow((sqrt(t_m) * (k * sin(k))), 2.0) / (l * cos(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 (k <= 5d-29) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = 2.0d0 / ((1.0d0 / l) * (((sqrt(t_m) * (k * sin(k))) ** 2.0d0) / (l * cos(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 (k <= 5e-29) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((1.0 / l) * (Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0) / (l * Math.cos(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 k <= 5e-29:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = 2.0 / ((1.0 / l) * (math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0) / (l * math.cos(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 (k <= 5e-29)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64((Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0) / Float64(l * cos(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 (k <= 5e-29)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = 2.0 / ((1.0 / l) * (((sqrt(t_m) * (k * sin(k))) ^ 2.0) / (l * cos(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[k, 5e-29], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999986e-29

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 30.2%

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

      2. pow2 30.2%

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

      3. *-commutative 30.2%

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

      4. sqrt-prod 16.6%

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

      5. sqrt-div 17.1%

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

      6. sqrt-pow1 19.6%

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

      7. metadata-eval 19.6%

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

      8. sqrt-prod 13.4%

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

      9. add-sqr-sqrt 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. *-commutative 26.8%

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

   8. Simplified 26.8%

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

   9. Taylor expanded in k around 0 23.6%

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

   if 4.99999999999999986e-29 < k

   1. Initial program 47.2%

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

   3. Simplified 47.2%

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

      1. associate-*l* 47.3%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.5%

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

      7. clear-num 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-/r/ 53.5%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 80.6%

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

       1. div-inv 80.6%

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

       2. add-sqr-sqrt 38.2%

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

       3. pow2 38.2%

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

       4. sqrt-prod 38.1%

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

       5. sqrt-pow1 43.7%

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

       6. metadata-eval 43.7%

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

       7. pow1 43.7%

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

       8. *-commutative 43.7%

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

       9. sqrt-prod 43.7%

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

       10. sqrt-pow1 43.7%

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

       11. metadata-eval 43.7%

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

       12. pow1 43.7%

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

   11. Applied egg-rr 43.7%

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

       1. associate-*r/ 43.7%

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

       2. *-commutative 43.7%

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

       3. *-rgt-identity 43.7%

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

       4. *-commutative 43.7%

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

       5. *-commutative 43.7%

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

       6. associate-*l* 43.7%

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

   13. Simplified 43.7%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot \left(\sqrt{t\_m} \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

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-28)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (/ 2.0 (/ (/ (pow (* k (* (sqrt t_m) (sin k))) 2.0) (* l (cos k))) l)))))

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-28) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((pow((k * (sqrt(t_m) * sin(k))), 2.0) / (l * cos(k))) / l);
	}
	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 <= 1.7d-28) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = 2.0d0 / ((((k * (sqrt(t_m) * sin(k))) ** 2.0d0) / (l * cos(k))) / l)
    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 <= 1.7e-28) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((Math.pow((k * (Math.sqrt(t_m) * Math.sin(k))), 2.0) / (l * Math.cos(k))) / l);
	}
	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 <= 1.7e-28:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = 2.0 / ((math.pow((k * (math.sqrt(t_m) * math.sin(k))), 2.0) / (l * math.cos(k))) / l)
	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-28)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k * Float64(sqrt(t_m) * sin(k))) ^ 2.0) / Float64(l * cos(k))) / l));
	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 <= 1.7e-28)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = 2.0 / ((((k * (sqrt(t_m) * sin(k))) ^ 2.0) / (l * cos(k))) / l);
	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, 1.7e-28], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e-28

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 30.2%

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

      2. pow2 30.2%

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

      3. *-commutative 30.2%

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

      4. sqrt-prod 16.6%

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

      5. sqrt-div 17.1%

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

      6. sqrt-pow1 19.6%

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

      7. metadata-eval 19.6%

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

      8. sqrt-prod 13.4%

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

      9. add-sqr-sqrt 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. *-commutative 26.8%

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

   8. Simplified 26.8%

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

   9. Taylor expanded in k around 0 23.6%

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

   if 1.7e-28 < k

   1. Initial program 47.2%

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

   3. Simplified 47.2%

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

      1. associate-*l* 47.3%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.5%

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

      7. clear-num 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-/r/ 53.5%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 80.6%

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

       1. associate-*l/ 80.6%

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

       2. *-un-lft-identity 80.6%

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

       3. add-sqr-sqrt 38.2%

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

       4. pow2 38.2%

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

       5. sqrt-prod 38.2%

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

       6. sqrt-pow1 43.8%

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

       7. metadata-eval 43.8%

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

       8. pow1 43.8%

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

       9. *-commutative 43.8%

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

       10. sqrt-prod 43.7%

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

       11. sqrt-pow1 43.7%

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

       12. metadata-eval 43.7%

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

       13. pow1 43.7%

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

   11. Applied egg-rr 43.7%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot \left(\sqrt{t} \cdot \sin k\right)\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

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 7.6e-29)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (/
     2.0
     (* (/ 1.0 l) (/ (* (pow k 2.0) (* t_m (pow k 2.0))) (* l (cos 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 (k <= 7.6e-29) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((1.0 / l) * ((pow(k, 2.0) * (t_m * pow(k, 2.0))) / (l * cos(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 (k <= 7.6d-29) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = 2.0d0 / ((1.0d0 / l) * (((k ** 2.0d0) * (t_m * (k ** 2.0d0))) / (l * cos(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 (k <= 7.6e-29) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = 2.0 / ((1.0 / l) * ((Math.pow(k, 2.0) * (t_m * Math.pow(k, 2.0))) / (l * Math.cos(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 k <= 7.6e-29:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = 2.0 / ((1.0 / l) * ((math.pow(k, 2.0) * (t_m * math.pow(k, 2.0))) / (l * math.cos(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 (k <= 7.6e-29)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64((k ^ 2.0) * Float64(t_m * (k ^ 2.0))) / Float64(l * cos(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 (k <= 7.6e-29)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = 2.0 / ((1.0 / l) * (((k ^ 2.0) * (t_m * (k ^ 2.0))) / (l * cos(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[k, 7.6e-29], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.59999999999999951e-29

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 30.2%

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

      2. pow2 30.2%

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

      3. *-commutative 30.2%

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

      4. sqrt-prod 16.6%

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

      5. sqrt-div 17.1%

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

      6. sqrt-pow1 19.6%

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

      7. metadata-eval 19.6%

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

      8. sqrt-prod 13.4%

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

      9. add-sqr-sqrt 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. *-commutative 26.8%

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

   8. Simplified 26.8%

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

   9. Taylor expanded in k around 0 23.6%

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

   if 7.59999999999999951e-29 < k

   1. Initial program 47.2%

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

   3. Simplified 47.2%

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

      1. associate-*l* 47.3%

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

      2. associate-/r* 52.3%

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

      3. associate-+r+ 52.3%

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

      4. metadata-eval 52.3%

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

      5. associate-*l* 52.3%

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

      6. associate-*l/ 53.5%

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

      7. clear-num 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-/r/ 53.5%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 80.6%

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

   10. Taylor expanded in k around 0 67.4%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 2.4e-14)
    (/ 2.0 (* (/ 1.0 l) (/ (* (pow k 2.0) (* t_m (pow k 2.0))) (* l (cos k)))))
    (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (pow 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 (t_m <= 2.4e-14) {
		tmp = 2.0 / ((1.0 / l) * ((pow(k, 2.0) * (t_m * pow(k, 2.0))) / (l * cos(k))));
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * pow(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 (t_m <= 2.4e-14) {
		tmp = 2.0 / ((1.0 / l) * ((Math.pow(k, 2.0) * (t_m * Math.pow(k, 2.0))) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * Math.pow(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 (t_m <= 2.4e-14)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64((k ^ 2.0) * Float64(t_m * (k ^ 2.0))) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * (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[t$95$m, 2.4e-14], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e-14

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 53.0%

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

      3. associate-+r+ 53.0%

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

      4. metadata-eval 53.0%

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

      5. associate-*l* 53.0%

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

      6. associate-*l/ 53.9%

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

      7. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-/r/ 53.9%

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

      2. associate-*l* 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in t around 0 76.9%

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

   10. Taylor expanded in k around 0 66.3%

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

   if 2.4e-14 < t

   1. Initial program 66.8%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 60.5%

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

      1. add-cube-cbrt 60.3%

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

      2. pow3 60.3%

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

      3. associate-/l/ 55.9%

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

      4. unpow2 55.9%

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

      5. cbrt-div 55.9%

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

      6. unpow3 55.9%

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

      7. add-cbrt-cube 57.8%

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

      8. unpow2 57.8%

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

      9. cbrt-prod 64.6%

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

      10. unpow2 64.6%

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

      11. div-inv 64.6%

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

      12. unpow-prod-down 55.9%

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

      13. pow-flip 55.9%

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

      14. metadata-eval 55.9%

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

   7. Applied egg-rr 55.9%

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

      1. cube-prod 64.6%

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

   9. Simplified 64.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.18 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{1.5} \cdot \left(\frac{1}{\ell} \cdot {t\_m}^{1.5}\right)}{\ell}}\\ \end{array} \end{array} \]

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 1.18e-14)
    (/ 2.0 (* (/ 1.0 l) (/ (* (pow k 2.0) (* t_m (pow k 2.0))) (* l (cos k)))))
    (/
     2.0
     (*
      (* 2.0 (pow k 2.0))
      (/ (* (pow t_m 1.5) (* (/ 1.0 l) (pow t_m 1.5))) l))))))

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 <= 1.18e-14) {
		tmp = 2.0 / ((1.0 / l) * ((pow(k, 2.0) * (t_m * pow(k, 2.0))) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 1.5) * ((1.0 / l) * pow(t_m, 1.5))) / l));
	}
	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 <= 1.18d-14) then
        tmp = 2.0d0 / ((1.0d0 / l) * (((k ** 2.0d0) * (t_m * (k ** 2.0d0))) / (l * cos(k))))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 1.5d0) * ((1.0d0 / l) * (t_m ** 1.5d0))) / l))
    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 <= 1.18e-14) {
		tmp = 2.0 / ((1.0 / l) * ((Math.pow(k, 2.0) * (t_m * Math.pow(k, 2.0))) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 1.5) * ((1.0 / l) * Math.pow(t_m, 1.5))) / l));
	}
	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 <= 1.18e-14:
		tmp = 2.0 / ((1.0 / l) * ((math.pow(k, 2.0) * (t_m * math.pow(k, 2.0))) / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 1.5) * ((1.0 / l) * math.pow(t_m, 1.5))) / l))
	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 <= 1.18e-14)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64((k ^ 2.0) * Float64(t_m * (k ^ 2.0))) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 1.5) * Float64(Float64(1.0 / l) * (t_m ^ 1.5))) / l)));
	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 <= 1.18e-14)
		tmp = 2.0 / ((1.0 / l) * (((k ^ 2.0) * (t_m * (k ^ 2.0))) / (l * cos(k))));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 1.5) * ((1.0 / l) * (t_m ^ 1.5))) / l));
	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, 1.18e-14], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[(1.0 / l), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.17999999999999993e-14

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 53.0%

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

      3. associate-+r+ 53.0%

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

      4. metadata-eval 53.0%

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

      5. associate-*l* 53.0%

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

      6. associate-*l/ 53.9%

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

      7. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-/r/ 53.9%

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

      2. associate-*l* 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in t around 0 76.9%

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

   10. Taylor expanded in k around 0 66.3%

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

   if 1.17999999999999993e-14 < t

   1. Initial program 66.8%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 60.5%

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

      1. div-inv 60.5%

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

      2. sqr-pow 60.4%

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

      3. associate-*l* 63.3%

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

      4. metadata-eval 63.3%

         \[\leadsto \frac{2}{\frac{{t}^{\color{blue}{1.5}} \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \frac{1}{\ell}\right)}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      5. metadata-eval 63.3%

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

   7. Applied egg-rr 63.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{1.5} \cdot \left({t}^{1.5} \cdot \frac{1}{\ell}\right)}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{1.5} \cdot \left(\frac{1}{\ell} \cdot {t}^{1.5}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 7e-15)
    (/ 2.0 (* (/ 1.0 l) (/ (* (pow k 2.0) (* t_m (pow k 2.0))) (* l (cos k)))))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))

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 <= 7e-15) {
		tmp = 2.0 / ((1.0 / l) * ((pow(k, 2.0) * (t_m * pow(k, 2.0))) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l));
	}
	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 <= 7e-15) {
		tmp = 2.0 / ((1.0 / l) * ((Math.pow(k, 2.0) * (t_m * Math.pow(k, 2.0))) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	}
	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 <= 7e-15)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64((k ^ 2.0) * Float64(t_m * (k ^ 2.0))) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	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, 7e-15], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.0000000000000001e-15

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 53.0%

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

      3. associate-+r+ 53.0%

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

      4. metadata-eval 53.0%

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

      5. associate-*l* 53.0%

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

      6. associate-*l/ 53.9%

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

      7. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-/r/ 53.9%

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

      2. associate-*l* 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in t around 0 76.9%

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

   10. Taylor expanded in k around 0 66.3%

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

   if 7.0000000000000001e-15 < t

   1. Initial program 66.8%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 60.5%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. cbrt-div 60.4%

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

      4. rem-cbrt-cube 63.2%

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

   7. Applied egg-rr 63.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{t\_m \cdot {k}^{4}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 7.1e-16)
    (/ 2.0 (* (/ 1.0 l) (/ (* t_m (pow k 4.0)) (* l (cos k)))))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))

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 <= 7.1e-16) {
		tmp = 2.0 / ((1.0 / l) * ((t_m * pow(k, 4.0)) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l));
	}
	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 <= 7.1e-16) {
		tmp = 2.0 / ((1.0 / l) * ((t_m * Math.pow(k, 4.0)) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	}
	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 <= 7.1e-16)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m * (k ^ 4.0)) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	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, 7.1e-16], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.1e-16

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 53.0%

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

      3. associate-+r+ 53.0%

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

      4. metadata-eval 53.0%

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

      5. associate-*l* 53.0%

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

      6. associate-*l/ 53.9%

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

      7. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-/r/ 53.9%

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

      2. associate-*l* 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in t around 0 76.9%

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

   10. Taylor expanded in k around 0 66.1%

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

   if 7.1e-16 < t

   1. Initial program 66.8%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 60.5%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. cbrt-div 60.4%

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

      4. rem-cbrt-cube 63.2%

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

   7. Applied egg-rr 63.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{t \cdot {k}^{4}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{t\_m \cdot {k}^{4}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{2} \cdot \left(t\_m \cdot \frac{1}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 5.35e-14)
    (/ 2.0 (* (/ 1.0 l) (/ (* t_m (pow k 4.0)) (* l (cos k)))))
    (/
     2.0
     (* (* 2.0 (pow k 2.0)) (/ (* (pow t_m 2.0) (* t_m (/ 1.0 l))) l))))))

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 <= 5.35e-14) {
		tmp = 2.0 / ((1.0 / l) * ((t_m * pow(k, 4.0)) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 2.0) * (t_m * (1.0 / l))) / l));
	}
	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 <= 5.35d-14) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((t_m * (k ** 4.0d0)) / (l * cos(k))))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 2.0d0) * (t_m * (1.0d0 / l))) / l))
    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 <= 5.35e-14) {
		tmp = 2.0 / ((1.0 / l) * ((t_m * Math.pow(k, 4.0)) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 2.0) * (t_m * (1.0 / l))) / l));
	}
	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 <= 5.35e-14:
		tmp = 2.0 / ((1.0 / l) * ((t_m * math.pow(k, 4.0)) / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 2.0) * (t_m * (1.0 / l))) / l))
	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 <= 5.35e-14)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m * (k ^ 4.0)) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 2.0) * Float64(t_m * Float64(1.0 / l))) / l)));
	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 <= 5.35e-14)
		tmp = 2.0 / ((1.0 / l) * ((t_m * (k ^ 4.0)) / (l * cos(k))));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 2.0) * (t_m * (1.0 / l))) / l));
	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, 5.35e-14], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3499999999999999e-14

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

      1. associate-*l* 45.8%

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

      2. associate-/r* 53.0%

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

      3. associate-+r+ 53.0%

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

      4. metadata-eval 53.0%

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

      5. associate-*l* 53.0%

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

      6. associate-*l/ 53.9%

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

      7. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-/r/ 53.9%

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

      2. associate-*l* 53.8%

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

   8. Simplified 53.8%

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

   9. Taylor expanded in t around 0 76.9%

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

   10. Taylor expanded in k around 0 66.1%

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

   if 5.3499999999999999e-14 < t

   1. Initial program 66.8%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 60.5%

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

      1. div-inv 60.5%

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

      2. unpow3 60.5%

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

      3. associate-*l* 62.0%

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

      4. pow2 62.0%

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

   7. Applied egg-rr 62.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{t \cdot {k}^{4}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \left(t \cdot \frac{1}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{t\_m \cdot {k}^{4}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 4.2e-20)
    (/ 2.0 (* (/ 1.0 l) (/ (* t_m (pow k 4.0)) (* l (cos k)))))
    (/ 2.0 (/ (* 2.0 (/ (* (pow k 2.0) (pow t_m 3.0)) l)) l)))))

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 <= 4.2e-20) {
		tmp = 2.0 / ((1.0 / l) * ((t_m * pow(k, 4.0)) / (l * cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * ((pow(k, 2.0) * pow(t_m, 3.0)) / l)) / l);
	}
	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 <= 4.2d-20) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((t_m * (k ** 4.0d0)) / (l * cos(k))))
    else
        tmp = 2.0d0 / ((2.0d0 * (((k ** 2.0d0) * (t_m ** 3.0d0)) / l)) / l)
    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 <= 4.2e-20) {
		tmp = 2.0 / ((1.0 / l) * ((t_m * Math.pow(k, 4.0)) / (l * Math.cos(k))));
	} else {
		tmp = 2.0 / ((2.0 * ((Math.pow(k, 2.0) * Math.pow(t_m, 3.0)) / l)) / l);
	}
	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 <= 4.2e-20:
		tmp = 2.0 / ((1.0 / l) * ((t_m * math.pow(k, 4.0)) / (l * math.cos(k))))
	else:
		tmp = 2.0 / ((2.0 * ((math.pow(k, 2.0) * math.pow(t_m, 3.0)) / l)) / l)
	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 <= 4.2e-20)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m * (k ^ 4.0)) / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(Float64((k ^ 2.0) * (t_m ^ 3.0)) / l)) / l));
	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 <= 4.2e-20)
		tmp = 2.0 / ((1.0 / l) * ((t_m * (k ^ 4.0)) / (l * cos(k))));
	else
		tmp = 2.0 / ((2.0 * (((k ^ 2.0) * (t_m ^ 3.0)) / l)) / l);
	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, 4.2e-20], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.1999999999999998e-20

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

      1. associate-*l* 45.7%

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

      2. associate-/r* 53.1%

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

      3. associate-+r+ 53.1%

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

      4. metadata-eval 53.1%

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

      5. associate-*l* 53.1%

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

      6. associate-*l/ 53.9%

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

      7. clear-num 53.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-/r/ 53.9%

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

      2. associate-*l* 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in t around 0 77.2%

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

   10. Taylor expanded in k around 0 66.3%

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

   if 4.1999999999999998e-20 < t

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

      1. associate-*l* 62.0%

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

      2. associate-/r* 67.5%

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

      3. associate-+r+ 67.5%

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

      4. metadata-eval 67.5%

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

      5. associate-*l* 67.5%

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

      6. associate-*l/ 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in k around 0 61.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \frac{t \cdot {k}^{4}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 4.6e-42)
    (/ 2.0 (* (/ 1.0 l) (* (/ t_m l) (pow k 4.0))))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (/ (pow t_m 3.0) l) l))))))

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 <= 4.6e-42) {
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 3.0) / l) / l));
	}
	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 <= 4.6d-42) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((t_m / l) * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 3.0d0) / l) / l))
    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 <= 4.6e-42) {
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 3.0) / l) / l));
	}
	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 <= 4.6e-42:
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 3.0) / l) / l))
	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 <= 4.6e-42)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m / l) * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / l)));
	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 <= 4.6e-42)
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * (k ^ 4.0)));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 3.0) / l) / l));
	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, 4.6e-42], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.60000000000000008e-42

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

      1. associate-*l* 45.0%

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

      2. associate-/r* 52.5%

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

      3. associate-+r+ 52.5%

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

      4. metadata-eval 52.5%

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

      5. associate-*l* 52.6%

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

      6. associate-*l/ 53.4%

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

      7. clear-num 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. associate-/r/ 53.4%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 77.2%

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

   10. Taylor expanded in k around 0 63.1%

       \[\leadsto \frac{2}{\frac{1}{\ell} \cdot \color{blue}{\frac{{k}^{4} \cdot t}{\ell}}} \]
   11. Step-by-step derivation

       1. associate-/l* 64.6%

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

   12. Simplified 64.6%

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

   if 4.60000000000000008e-42 < t

   1. Initial program 66.9%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 59.7%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\frac{t}{\ell} \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 4.6e-24)
    (/ 2.0 (* (/ 1.0 l) (* (/ t_m l) (pow k 4.0))))
    (/ 2.0 (/ (* 2.0 (/ (* (pow k 2.0) (pow t_m 3.0)) l)) l)))))

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 <= 4.6e-24) {
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * ((pow(k, 2.0) * pow(t_m, 3.0)) / l)) / l);
	}
	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 <= 4.6d-24) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((t_m / l) * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * (((k ** 2.0d0) * (t_m ** 3.0d0)) / l)) / l)
    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 <= 4.6e-24) {
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * ((Math.pow(k, 2.0) * Math.pow(t_m, 3.0)) / l)) / l);
	}
	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 <= 4.6e-24:
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((2.0 * ((math.pow(k, 2.0) * math.pow(t_m, 3.0)) / l)) / l)
	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 <= 4.6e-24)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m / l) * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(Float64((k ^ 2.0) * (t_m ^ 3.0)) / l)) / l));
	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 <= 4.6e-24)
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * (k ^ 4.0)));
	else
		tmp = 2.0 / ((2.0 * (((k ^ 2.0) * (t_m ^ 3.0)) / l)) / l);
	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, 4.6e-24], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.6000000000000002e-24

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

      1. associate-*l* 45.1%

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

      2. associate-/r* 52.5%

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

      3. associate-+r+ 52.5%

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

      4. metadata-eval 52.5%

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

      5. associate-*l* 52.6%

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

      6. associate-*l/ 53.4%

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

      7. clear-num 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. associate-/r/ 53.4%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 77.0%

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

   10. Taylor expanded in k around 0 63.0%

       \[\leadsto \frac{2}{\frac{1}{\ell} \cdot \color{blue}{\frac{{k}^{4} \cdot t}{\ell}}} \]
   11. Step-by-step derivation

       1. associate-/l* 64.4%

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

   12. Simplified 64.4%

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

   if 4.6000000000000002e-24 < t

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

      1. associate-*l* 63.0%

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

      2. associate-/r* 68.4%

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

      3. associate-+r+ 68.4%

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

      4. metadata-eval 68.4%

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

      5. associate-*l* 68.4%

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

      6. associate-*l/ 71.0%

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

   6. Applied egg-rr 71.0%

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

   7. Taylor expanded in k around 0 61.3%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\frac{t}{\ell} \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 3.7e-43)
    (/ 2.0 (* (/ 1.0 l) (* (/ t_m l) (pow k 4.0))))
    (/ 2.0 (/ (* (* 2.0 (pow k 2.0)) (/ (pow t_m 3.0) l)) l)))))

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 <= 3.7e-43) {
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * pow(k, 4.0)));
	} else {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	}
	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 <= 3.7d-43) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((t_m / l) * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    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 <= 3.7e-43) {
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	}
	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 <= 3.7e-43:
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	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 <= 3.7e-43)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m / l) * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	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 <= 3.7e-43)
		tmp = 2.0 / ((1.0 / l) * ((t_m / l) * (k ^ 4.0)));
	else
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	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, 3.7e-43], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.7e-43

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

      1. associate-*l* 45.0%

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

      2. associate-/r* 52.5%

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

      3. associate-+r+ 52.5%

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

      4. metadata-eval 52.5%

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

      5. associate-*l* 52.6%

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

      6. associate-*l/ 53.4%

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

      7. clear-num 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. associate-/r/ 53.4%

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

      2. associate-*l* 53.4%

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

   8. Simplified 53.4%

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

   9. Taylor expanded in t around 0 77.2%

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

   10. Taylor expanded in k around 0 63.1%

       \[\leadsto \frac{2}{\frac{1}{\ell} \cdot \color{blue}{\frac{{k}^{4} \cdot t}{\ell}}} \]
   11. Step-by-step derivation

       1. associate-/l* 64.6%

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

   12. Simplified 64.6%

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

   if 3.7e-43 < t

   1. Initial program 66.9%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 59.7%

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

      1. associate-*l/ 61.1%

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

   7. Applied egg-rr 61.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\frac{t}{\ell} \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 56.4% accurate, 3.8× speedup

Math

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

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) (* (/ t_m l) (pow k 4.0))))))

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 * (2.0 / ((1.0 / l) * ((t_m / l) * pow(k, 4.0))));
}

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 * (2.0d0 / ((1.0d0 / l) * ((t_m / l) * (k ** 4.0d0))))
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 * (2.0 / ((1.0 / l) * ((t_m / l) * Math.pow(k, 4.0))));
}

Python

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

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(t_m / l) * (k ^ 4.0)))))
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 * (2.0 / ((1.0 / l) * ((t_m / l) * (k ^ 4.0))));
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[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Simplified 52.9%

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

   1. associate-*l* 50.4%

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

   2. associate-/r* 57.2%

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

   3. associate-+r+ 57.2%

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

   4. metadata-eval 57.2%

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

   5. associate-*l* 57.2%

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

   6. associate-*l/ 58.5%

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

   7. clear-num 58.5%

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

6. Applied egg-rr 58.5%

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

   1. associate-/r/ 58.5%

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

   2. associate-*l* 58.5%

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

8. Simplified 58.5%

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

9. Taylor expanded in t around 0 69.9%

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

10. Taylor expanded in k around 0 58.5%

    \[\leadsto \frac{2}{\frac{1}{\ell} \cdot \color{blue}{\frac{{k}^{4} \cdot t}{\ell}}} \]
11. Step-by-step derivation

    1. associate-/l* 59.4%

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

12. Simplified 59.4%

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

13. Final simplification 59.4%

    \[\leadsto \frac{2}{\frac{1}{\ell} \cdot \left(\frac{t}{\ell} \cdot {k}^{4}\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))