Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.2% → 90.2%

Time: 26.9s

Alternatives: 18

Speedup: 3.8×

• 27.1% 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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.58 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{{t_m}^{0.75}}{\frac{\frac{\ell}{{t_m}^{0.75}}}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.58e-55)
    (/
     2.0
     (* (* (* (/ k l) (/ 1.0 l)) (/ k (cos k))) (* t_m (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (/ (pow (/ (pow t_m 0.75) (/ (/ l (pow t_m 0.75)) (sin k))) 2.0) (cos k))
      (+ 2.0 (pow (/ k t_m) 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.58e-55) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((pow((pow(t_m, 0.75) / ((l / pow(t_m, 0.75)) / sin(k))), 2.0) / cos(k)) * (2.0 + pow((k / t_m), 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 <= 1.58d-55) then
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / (((((t_m ** 0.75d0) / ((l / (t_m ** 0.75d0)) / sin(k))) ** 2.0d0) / cos(k)) * (2.0d0 + ((k / t_m) ** 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 <= 1.58e-55) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((Math.pow((Math.pow(t_m, 0.75) / ((l / Math.pow(t_m, 0.75)) / Math.sin(k))), 2.0) / Math.cos(k)) * (2.0 + Math.pow((k / t_m), 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 <= 1.58e-55:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((math.pow((math.pow(t_m, 0.75) / ((l / math.pow(t_m, 0.75)) / math.sin(k))), 2.0) / math.cos(k)) * (2.0 + math.pow((k / t_m), 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.58e-55)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64((t_m ^ 0.75) / Float64(Float64(l / (t_m ^ 0.75)) / sin(k))) ^ 2.0) / cos(k)) * Float64(2.0 + (Float64(k / t_m) ^ 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 <= 1.58e-55)
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / (((((t_m ^ 0.75) / ((l / (t_m ^ 0.75)) / sin(k))) ^ 2.0) / cos(k)) * (2.0 + ((k / t_m) ^ 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, 1.58e-55], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] / N[(N[(l / N[Power[t$95$m, 0.75], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.58000000000000007e-55

   1. Initial program 49.6%

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

   2. Taylor expanded in t around 0 66.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.3%

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

   4. Simplified 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.9%

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

      5. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 1.58000000000000007e-55 < t

   1. Initial program 63.8%

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

   3. Simplified 72.2%

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

      1. associate-/l/ 64.8%

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

      2. associate-*l/ 63.8%

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

      3. add-sqr-sqrt 63.7%

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

      4. associate-*l* 63.7%

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

      5. sqrt-div 63.7%

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

      6. sqrt-pow1 63.7%

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

      7. sqrt-prod 28.8%

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

      8. add-sqr-sqrt 53.3%

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

      9. metadata-eval 53.3%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 55.7%

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

      12. sqrt-prod 39.1%

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

      13. add-sqr-sqrt 82.0%

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

      14. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. tan-quot 82.0%

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

      2. associate-*r/ 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r* 85.1%

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

      5. pow1 85.1%

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

      6. metadata-eval 85.1%

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

      7. pow1 85.1%

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

      8. metadata-eval 85.1%

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

      9. pow-sqr 85.1%

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

      10. associate-*l/ 85.1%

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

      11. associate-/l* 81.7%

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

      12. metadata-eval 81.7%

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

      13. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. sqr-pow 81.6%

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

      2. div-inv 81.6%

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

      3. times-frac 94.8%

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

      4. metadata-eval 94.8%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{\color{blue}{0.75}}}{\ell} \cdot \frac{{t}^{\left(\frac{1.5}{2}\right)}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. metadata-eval 94.8%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{\color{blue}{0.75}}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   9. Applied egg-rr 94.8%

      \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{0.75}}{\frac{1}{\sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-/r/ 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \color{blue}{\left(\frac{{t}^{0.75}}{1} \cdot \sin k\right)}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

       2. /-rgt-identity 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left(\color{blue}{{t}^{0.75}} \cdot \sin k\right)\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   11. Simplified 94.9%

       \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \left({t}^{0.75} \cdot \sin k\right)\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 92.6%

          \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75} \cdot \left({t}^{0.75} \cdot \sin k\right)}{\ell}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

       2. associate-/l* 94.8%

          \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\frac{\ell}{{t}^{0.75} \cdot \sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   13. Applied egg-rr 94.8%

       \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\frac{\ell}{{t}^{0.75} \cdot \sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   14. Step-by-step derivation

       1. associate-/r* 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\color{blue}{\frac{\frac{\ell}{{t}^{0.75}}}{\sin k}}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   15. Simplified 94.9%

       \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\frac{\frac{\ell}{{t}^{0.75}}}{\sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.58 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\frac{\frac{\ell}{{t}^{0.75}}}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]

Alternative 2: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right) \cdot \frac{{\left(\frac{{t_m}^{0.75}}{\ell} \cdot \left(\sin k \cdot {t_m}^{0.75}\right)\right)}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.1e-55)
    (/
     2.0
     (* (* (* (/ k l) (/ 1.0 l)) (/ k (cos k))) (* t_m (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (+ 2.0 (pow (/ k t_m) 2.0))
      (/
       (pow (* (/ (pow t_m 0.75) l) (* (sin k) (pow t_m 0.75))) 2.0)
       (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 (t_m <= 6.1e-55) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((2.0 + pow((k / t_m), 2.0)) * (pow(((pow(t_m, 0.75) / l) * (sin(k) * pow(t_m, 0.75))), 2.0) / 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 (t_m <= 6.1d-55) then
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 + ((k / t_m) ** 2.0d0)) * (((((t_m ** 0.75d0) / l) * (sin(k) * (t_m ** 0.75d0))) ** 2.0d0) / 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 (t_m <= 6.1e-55) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.pow(((Math.pow(t_m, 0.75) / l) * (Math.sin(k) * Math.pow(t_m, 0.75))), 2.0) / 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 t_m <= 6.1e-55:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((2.0 + math.pow((k / t_m), 2.0)) * (math.pow(((math.pow(t_m, 0.75) / l) * (math.sin(k) * math.pow(t_m, 0.75))), 2.0) / 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 (t_m <= 6.1e-55)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64((Float64(Float64((t_m ^ 0.75) / l) * Float64(sin(k) * (t_m ^ 0.75))) ^ 2.0) / 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 (t_m <= 6.1e-55)
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / ((2.0 + ((k / t_m) ^ 2.0)) * (((((t_m ^ 0.75) / l) * (sin(k) * (t_m ^ 0.75))) ^ 2.0) / 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[t$95$m, 6.1e-55], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[(N[Power[t$95$m, 0.75], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 0.75], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.1000000000000001e-55

   1. Initial program 49.6%

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

   2. Taylor expanded in t around 0 66.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.3%

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

   4. Simplified 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.9%

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

      5. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 6.1000000000000001e-55 < t

   1. Initial program 63.8%

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

   3. Simplified 72.2%

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

      1. associate-/l/ 64.8%

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

      2. associate-*l/ 63.8%

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

      3. add-sqr-sqrt 63.7%

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

      4. associate-*l* 63.7%

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

      5. sqrt-div 63.7%

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

      6. sqrt-pow1 63.7%

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

      7. sqrt-prod 28.8%

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

      8. add-sqr-sqrt 53.3%

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

      9. metadata-eval 53.3%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 55.7%

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

      12. sqrt-prod 39.1%

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

      13. add-sqr-sqrt 82.0%

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

      14. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. tan-quot 82.0%

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

      2. associate-*r/ 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r* 85.1%

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

      5. pow1 85.1%

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

      6. metadata-eval 85.1%

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

      7. pow1 85.1%

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

      8. metadata-eval 85.1%

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

      9. pow-sqr 85.1%

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

      10. associate-*l/ 85.1%

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

      11. associate-/l* 81.7%

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

      12. metadata-eval 81.7%

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

      13. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. sqr-pow 81.6%

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

      2. div-inv 81.6%

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

      3. times-frac 94.8%

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

      4. metadata-eval 94.8%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{\color{blue}{0.75}}}{\ell} \cdot \frac{{t}^{\left(\frac{1.5}{2}\right)}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. metadata-eval 94.8%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{\color{blue}{0.75}}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   9. Applied egg-rr 94.8%

      \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{0.75}}{\frac{1}{\sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-/r/ 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \color{blue}{\left(\frac{{t}^{0.75}}{1} \cdot \sin k\right)}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

       2. /-rgt-identity 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left(\color{blue}{{t}^{0.75}} \cdot \sin k\right)\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   11. Simplified 94.9%

       \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \left({t}^{0.75} \cdot \sin k\right)\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left(\sin k \cdot {t}^{0.75}\right)\right)}^{2}}{\cos k}}\\ \end{array} \]

Alternative 3: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 8.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\cos k} \cdot {\left(\sin k \cdot \frac{{t_m}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{{\left(\frac{{t_m}^{0.75}}{\ell} \cdot \left(k \cdot {t_m}^{0.75}\right)\right)}^{2}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 3.5e-57)
      (/
       2.0
       (* (* (* (/ k l) (/ 1.0 l)) (/ k (cos k))) (* t_m (pow (sin k) 2.0))))
      (if (<= t_m 8.5e+203)
        (/ 2.0 (* (/ t_2 (cos k)) (pow (* (sin k) (/ (pow t_m 1.5) l)) 2.0)))
        (/
         2.0
         (*
          t_2
          (/
           (pow (* (/ (pow t_m 0.75) l) (* k (pow t_m 0.75))) 2.0)
           (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 t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.5e-57) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	} else if (t_m <= 8.5e+203) {
		tmp = 2.0 / ((t_2 / cos(k)) * pow((sin(k) * (pow(t_m, 1.5) / l)), 2.0));
	} else {
		tmp = 2.0 / (t_2 * (pow(((pow(t_m, 0.75) / l) * (k * pow(t_m, 0.75))), 2.0) / 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) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 + ((k / t_m) ** 2.0d0)
    if (t_m <= 3.5d-57) then
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    else if (t_m <= 8.5d+203) then
        tmp = 2.0d0 / ((t_2 / cos(k)) * ((sin(k) * ((t_m ** 1.5d0) / l)) ** 2.0d0))
    else
        tmp = 2.0d0 / (t_2 * (((((t_m ** 0.75d0) / l) * (k * (t_m ** 0.75d0))) ** 2.0d0) / 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 t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.5e-57) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 8.5e+203) {
		tmp = 2.0 / ((t_2 / Math.cos(k)) * Math.pow((Math.sin(k) * (Math.pow(t_m, 1.5) / l)), 2.0));
	} else {
		tmp = 2.0 / (t_2 * (Math.pow(((Math.pow(t_m, 0.75) / l) * (k * Math.pow(t_m, 0.75))), 2.0) / 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):
	t_2 = 2.0 + math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 3.5e-57:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	elif t_m <= 8.5e+203:
		tmp = 2.0 / ((t_2 / math.cos(k)) * math.pow((math.sin(k) * (math.pow(t_m, 1.5) / l)), 2.0))
	else:
		tmp = 2.0 / (t_2 * (math.pow(((math.pow(t_m, 0.75) / l) * (k * math.pow(t_m, 0.75))), 2.0) / 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)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.5e-57)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	elseif (t_m <= 8.5e+203)
		tmp = Float64(2.0 / Float64(Float64(t_2 / cos(k)) * (Float64(sin(k) * Float64((t_m ^ 1.5) / l)) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64((Float64(Float64((t_m ^ 0.75) / l) * Float64(k * (t_m ^ 0.75))) ^ 2.0) / 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)
	t_2 = 2.0 + ((k / t_m) ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.5e-57)
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	elseif (t_m <= 8.5e+203)
		tmp = 2.0 / ((t_2 / cos(k)) * ((sin(k) * ((t_m ^ 1.5) / l)) ^ 2.0));
	else
		tmp = 2.0 / (t_2 * (((((t_m ^ 0.75) / l) * (k * (t_m ^ 0.75))) ^ 2.0) / 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_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-57], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+203], N[(2.0 / N[(N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Power[N[(N[(N[Power[t$95$m, 0.75], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Power[t$95$m, 0.75], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.49999999999999991e-57

   1. Initial program 49.6%

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

   2. Taylor expanded in t around 0 66.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.3%

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

   4. Simplified 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.9%

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

      5. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 3.49999999999999991e-57 < t < 8.50000000000000025e203

   1. Initial program 67.8%

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

   3. Simplified 74.5%

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

      1. associate-/l/ 69.3%

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

      2. associate-*l/ 67.8%

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

      3. add-sqr-sqrt 67.7%

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

      4. associate-*l* 67.7%

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

      5. sqrt-div 67.7%

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

      6. sqrt-pow1 67.7%

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

      7. sqrt-prod 34.4%

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

      8. add-sqr-sqrt 52.9%

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

      9. metadata-eval 52.9%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 56.2%

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

      12. sqrt-prod 47.3%

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

      13. add-sqr-sqrt 88.5%

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

      14. metadata-eval 88.5%

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

   5. Applied egg-rr 88.5%

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

      1. tan-quot 88.5%

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

      2. associate-*r/ 88.5%

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

      3. *-commutative 88.5%

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

      4. associate-*r* 93.0%

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

      5. pow1 93.0%

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

      6. metadata-eval 93.0%

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

      7. pow1 93.0%

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

      8. metadata-eval 93.0%

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

      9. pow-sqr 93.0%

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

      10. associate-*l/ 92.9%

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

      11. associate-/l* 88.5%

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

      12. metadata-eval 88.5%

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

      13. metadata-eval 88.5%

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

   7. Applied egg-rr 88.5%

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

      1. *-commutative 88.5%

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

      2. clear-num 88.5%

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

      3. un-div-inv 88.5%

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

      4. div-inv 88.5%

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

      5. clear-num 89.3%

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

   9. Applied egg-rr 89.3%

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

       1. associate-/r/ 89.2%

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

       2. associate-*r/ 92.9%

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

       3. *-commutative 92.9%

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

       4. associate-*r/ 93.1%

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

   11. Simplified 93.1%

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

   if 8.50000000000000025e203 < t

   1. Initial program 54.4%

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

   3. Simplified 66.7%

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

      1. associate-/l/ 54.4%

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

      2. associate-*l/ 54.4%

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

      3. add-sqr-sqrt 54.4%

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

      4. associate-*l* 54.4%

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

      5. sqrt-div 54.4%

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

      6. sqrt-pow1 54.4%

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

      7. sqrt-prod 15.6%

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

      8. add-sqr-sqrt 54.4%

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

      9. metadata-eval 54.4%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 54.4%

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

      12. sqrt-prod 19.9%

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

      13. add-sqr-sqrt 66.7%

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

      14. metadata-eval 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. tan-quot 66.7%

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

      2. associate-*r/ 66.7%

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

      3. *-commutative 66.7%

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

      4. associate-*r* 66.7%

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

      5. pow1 66.7%

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

      6. metadata-eval 66.7%

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

      7. pow1 66.7%

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

      8. metadata-eval 66.7%

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

      9. pow-sqr 66.7%

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

      10. associate-*l/ 66.7%

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

      11. associate-/l* 65.7%

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

      12. metadata-eval 65.7%

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

      13. metadata-eval 65.7%

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

   7. Applied egg-rr 65.7%

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

      1. sqr-pow 65.7%

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

      2. div-inv 65.7%

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

      3. times-frac 99.4%

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

      4. metadata-eval 99.4%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{\color{blue}{0.75}}}{\ell} \cdot \frac{{t}^{\left(\frac{1.5}{2}\right)}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{\color{blue}{0.75}}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   9. Applied egg-rr 99.4%

      \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{0.75}}{\frac{1}{\sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-/r/ 99.3%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \color{blue}{\left(\frac{{t}^{0.75}}{1} \cdot \sin k\right)}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

       2. /-rgt-identity 99.3%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left(\color{blue}{{t}^{0.75}} \cdot \sin k\right)\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   11. Simplified 99.3%

       \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \left({t}^{0.75} \cdot \sin k\right)\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   12. Taylor expanded in k around 0 99.3%

       \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left({t}^{0.75} \cdot \color{blue}{k}\right)\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\cos k} \cdot {\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left(k \cdot {t}^{0.75}\right)\right)}^{2}}{\cos k}}\\ \end{array} \]

Alternative 4: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{-2}{{\left(\frac{\sin k}{\frac{\ell}{{t_m}^{1.5}}}\right)}^{2} \cdot \left(-2 - {\left(\frac{k}{t_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-55)
    (/
     2.0
     (* (* (* (/ k l) (/ 1.0 l)) (/ k (cos k))) (* t_m (pow (sin k) 2.0))))
    (*
     (cos k)
     (/
      -2.0
      (*
       (pow (/ (sin k) (/ l (pow t_m 1.5))) 2.0)
       (- -2.0 (pow (/ k t_m) 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-55) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	} else {
		tmp = cos(k) * (-2.0 / (pow((sin(k) / (l / pow(t_m, 1.5))), 2.0) * (-2.0 - pow((k / t_m), 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 <= 3d-55) then
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    else
        tmp = cos(k) * ((-2.0d0) / (((sin(k) / (l / (t_m ** 1.5d0))) ** 2.0d0) * ((-2.0d0) - ((k / t_m) ** 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 <= 3e-55) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.cos(k) * (-2.0 / (Math.pow((Math.sin(k) / (l / Math.pow(t_m, 1.5))), 2.0) * (-2.0 - Math.pow((k / t_m), 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 <= 3e-55:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	else:
		tmp = math.cos(k) * (-2.0 / (math.pow((math.sin(k) / (l / math.pow(t_m, 1.5))), 2.0) * (-2.0 - math.pow((k / t_m), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-55)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	else
		tmp = Float64(cos(k) * Float64(-2.0 / Float64((Float64(sin(k) / Float64(l / (t_m ^ 1.5))) ^ 2.0) * Float64(-2.0 - (Float64(k / t_m) ^ 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 <= 3e-55)
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	else
		tmp = cos(k) * (-2.0 / (((sin(k) / (l / (t_m ^ 1.5))) ^ 2.0) * (-2.0 - ((k / t_m) ^ 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, 3e-55], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[k], $MachinePrecision] * N[(-2.0 / N[(N[Power[N[(N[Sin[k], $MachinePrecision] / N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-2.0 - N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000016e-55

   1. Initial program 49.6%

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

   2. Taylor expanded in t around 0 66.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.3%

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

   4. Simplified 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.9%

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

      5. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 3.00000000000000016e-55 < t

   1. Initial program 63.8%

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

   3. Simplified 72.2%

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

      1. associate-/l/ 64.8%

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

      2. associate-*l/ 63.8%

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

      3. add-sqr-sqrt 63.7%

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

      4. associate-*l* 63.7%

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

      5. sqrt-div 63.7%

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

      6. sqrt-pow1 63.7%

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

      7. sqrt-prod 28.8%

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

      8. add-sqr-sqrt 53.3%

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

      9. metadata-eval 53.3%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 55.7%

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

      12. sqrt-prod 39.1%

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

      13. add-sqr-sqrt 82.0%

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

      14. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. tan-quot 82.0%

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

      2. associate-*r/ 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r* 85.1%

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

      5. pow1 85.1%

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

      6. metadata-eval 85.1%

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

      7. pow1 85.1%

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

      8. metadata-eval 85.1%

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

      9. pow-sqr 85.1%

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

      10. associate-*l/ 85.1%

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

      11. associate-/l* 81.7%

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

      12. metadata-eval 81.7%

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

      13. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. sqr-pow 81.6%

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

      2. div-inv 81.6%

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

      3. times-frac 94.8%

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

      4. metadata-eval 94.8%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{\color{blue}{0.75}}}{\ell} \cdot \frac{{t}^{\left(\frac{1.5}{2}\right)}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. metadata-eval 94.8%

         \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{\color{blue}{0.75}}}{\frac{1}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   9. Applied egg-rr 94.8%

      \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{0.75}}{\frac{1}{\sin k}}\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-/r/ 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \color{blue}{\left(\frac{{t}^{0.75}}{1} \cdot \sin k\right)}\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

       2. /-rgt-identity 94.9%

          \[\leadsto \frac{2}{\frac{{\left(\frac{{t}^{0.75}}{\ell} \cdot \left(\color{blue}{{t}^{0.75}} \cdot \sin k\right)\right)}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   11. Simplified 94.9%

       \[\leadsto \frac{2}{\frac{{\color{blue}{\left(\frac{{t}^{0.75}}{\ell} \cdot \left({t}^{0.75} \cdot \sin k\right)\right)}}^{2}}{\cos k} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   13. Applied egg-rr 82.2%

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

       1. associate-*r/ 82.2%

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

       2. metadata-eval 82.2%

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

       3. associate-*l/ 82.2%

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

       4. associate-/r/ 82.2%

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

       5. *-commutative 82.2%

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

       6. associate-/r/ 85.0%

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

       7. unsub-neg 85.0%

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

   15. Simplified 85.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{-2}{{\left(\frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)}^{2} \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]

Alternative 5: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 + {\left(\frac{k}{t_m}\right)}^{2}}{\cos k} \cdot {\left(\sin k \cdot \frac{{t_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-57)
    (/
     2.0
     (* (* (* (/ k l) (/ 1.0 l)) (/ k (cos k))) (* t_m (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (/ (+ 2.0 (pow (/ k t_m) 2.0)) (cos k))
      (pow (* (sin k) (/ (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 <= 2.8e-57) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / (((2.0 + pow((k / t_m), 2.0)) / cos(k)) * pow((sin(k) * (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 <= 2.8d-57) then
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / (((2.0d0 + ((k / t_m) ** 2.0d0)) / cos(k)) * ((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 <= 2.8e-57) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / (((2.0 + Math.pow((k / t_m), 2.0)) / Math.cos(k)) * Math.pow((Math.sin(k) * (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 <= 2.8e-57:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / (((2.0 + math.pow((k / t_m), 2.0)) / math.cos(k)) * math.pow((math.sin(k) * (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 <= 2.8e-57)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) / cos(k)) * (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 <= 2.8e-57)
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / (((2.0 + ((k / t_m) ^ 2.0)) / cos(k)) * ((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, 2.8e-57], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.7999999999999999e-57

   1. Initial program 49.6%

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

   2. Taylor expanded in t around 0 66.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.3%

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

   4. Simplified 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.9%

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

      5. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 2.7999999999999999e-57 < t

   1. Initial program 63.8%

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

   3. Simplified 72.2%

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

      1. associate-/l/ 64.8%

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

      2. associate-*l/ 63.8%

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

      3. add-sqr-sqrt 63.7%

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

      4. associate-*l* 63.7%

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

      5. sqrt-div 63.7%

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

      6. sqrt-pow1 63.7%

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

      7. sqrt-prod 28.8%

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

      8. add-sqr-sqrt 53.3%

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

      9. metadata-eval 53.3%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 55.7%

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

      12. sqrt-prod 39.1%

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

      13. add-sqr-sqrt 82.0%

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

      14. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. tan-quot 82.0%

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

      2. associate-*r/ 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r* 85.1%

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

      5. pow1 85.1%

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

      6. metadata-eval 85.1%

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

      7. pow1 85.1%

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

      8. metadata-eval 85.1%

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

      9. pow-sqr 85.1%

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

      10. associate-*l/ 85.1%

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

      11. associate-/l* 81.7%

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

      12. metadata-eval 81.7%

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

      13. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. *-commutative 81.7%

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

      2. clear-num 81.7%

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

      3. un-div-inv 81.7%

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

      4. div-inv 81.7%

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

      5. clear-num 82.3%

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

   9. Applied egg-rr 82.3%

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

       1. associate-/r/ 82.2%

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

       2. associate-*r/ 85.1%

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

       3. *-commutative 85.1%

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

       4. associate-*r/ 85.2%

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

   11. Simplified 85.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\cos k} \cdot {\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{{t_m}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + \frac{\frac{k}{t_m}}{\frac{t_m}{k}}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.5e-56)
    (/
     2.0
     (* (* (* (/ k l) (/ 1.0 l)) (/ k (cos k))) (* t_m (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (/ (pow (/ (pow t_m 1.5) (/ l (sin k))) 2.0) (cos k))
      (+ 2.0 (/ (/ k t_m) (/ t_m 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 <= 2.5e-56) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((pow((pow(t_m, 1.5) / (l / sin(k))), 2.0) / cos(k)) * (2.0 + ((k / t_m) / (t_m / 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 (t_m <= 2.5d-56) then
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / (((((t_m ** 1.5d0) / (l / sin(k))) ** 2.0d0) / cos(k)) * (2.0d0 + ((k / t_m) / (t_m / 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 (t_m <= 2.5e-56) {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((Math.pow((Math.pow(t_m, 1.5) / (l / Math.sin(k))), 2.0) / Math.cos(k)) * (2.0 + ((k / t_m) / (t_m / 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 t_m <= 2.5e-56:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((math.pow((math.pow(t_m, 1.5) / (l / math.sin(k))), 2.0) / math.cos(k)) * (2.0 + ((k / t_m) / (t_m / 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 <= 2.5e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64((t_m ^ 1.5) / Float64(l / sin(k))) ^ 2.0) / cos(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / 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 (t_m <= 2.5e-56)
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / (((((t_m ^ 1.5) / (l / sin(k))) ^ 2.0) / cos(k)) * (2.0 + ((k / t_m) / (t_m / 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[t$95$m, 2.5e-56], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.49999999999999999e-56

   1. Initial program 49.6%

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

   2. Taylor expanded in t around 0 66.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.3%

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

   4. Simplified 66.3%

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

      1. unpow2 66.3%

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

      2. times-frac 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. metadata-eval 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.9%

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

      5. metadata-eval 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

   10. Simplified 75.9%

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

   if 2.49999999999999999e-56 < t

   1. Initial program 63.8%

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

   3. Simplified 72.2%

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

      1. associate-/l/ 64.8%

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

      2. associate-*l/ 63.8%

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

      3. add-sqr-sqrt 63.7%

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

      4. associate-*l* 63.7%

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

      5. sqrt-div 63.7%

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

      6. sqrt-pow1 63.7%

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

      7. sqrt-prod 28.8%

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

      8. add-sqr-sqrt 53.3%

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

      9. metadata-eval 53.3%

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

      10. sqrt-div 54.4%

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

      11. sqrt-pow1 55.7%

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

      12. sqrt-prod 39.1%

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

      13. add-sqr-sqrt 82.0%

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

      14. metadata-eval 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. tan-quot 82.0%

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

      2. associate-*r/ 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r* 85.1%

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

      5. pow1 85.1%

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

      6. metadata-eval 85.1%

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

      7. pow1 85.1%

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

      8. metadata-eval 85.1%

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

      9. pow-sqr 85.1%

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

      10. associate-*l/ 85.1%

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

      11. associate-/l* 81.7%

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

      12. metadata-eval 81.7%

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

      13. metadata-eval 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. unpow2 81.7%

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

      2. clear-num 81.6%

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

      3. un-div-inv 81.7%

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

   9. Applied egg-rr 81.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{{t}^{1.5}}{\frac{\ell}{\sin k}}\right)}^{2}}{\cos k} \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)}\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot {t_m}^{1.5}}\\ t_3 := \frac{k}{\cos k}\\ t_4 := t_m \cdot {\sin k}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.55:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{elif}\;k \leq 10^{+185}:\\ \;\;\;\;\frac{2}{\frac{-k}{\frac{-{\ell}^{2}}{t_3 \cdot t_4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot t_3\right) \cdot t_4}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* k (pow t_m 1.5))))
        (t_3 (/ k (cos k)))
        (t_4 (* t_m (pow (sin k) 2.0))))
   (*
    t_s
    (if (<= k 0.55)
      (* t_2 t_2)
      (if (<= k 1e+185)
        (/ 2.0 (/ (- k) (/ (- (pow l 2.0)) (* t_3 t_4))))
        (/ 2.0 (* (* (* (/ k l) (/ 1.0 l)) t_3) t_4)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * pow(t_m, 1.5));
	double t_3 = k / cos(k);
	double t_4 = t_m * pow(sin(k), 2.0);
	double tmp;
	if (k <= 0.55) {
		tmp = t_2 * t_2;
	} else if (k <= 1e+185) {
		tmp = 2.0 / (-k / (-pow(l, 2.0) / (t_3 * t_4)));
	} else {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_3) * t_4);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = l / (k * (t_m ** 1.5d0))
    t_3 = k / cos(k)
    t_4 = t_m * (sin(k) ** 2.0d0)
    if (k <= 0.55d0) then
        tmp = t_2 * t_2
    else if (k <= 1d+185) then
        tmp = 2.0d0 / (-k / (-(l ** 2.0d0) / (t_3 * t_4)))
    else
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * t_3) * t_4)
    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 t_2 = l / (k * Math.pow(t_m, 1.5));
	double t_3 = k / Math.cos(k);
	double t_4 = t_m * Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 0.55) {
		tmp = t_2 * t_2;
	} else if (k <= 1e+185) {
		tmp = 2.0 / (-k / (-Math.pow(l, 2.0) / (t_3 * t_4)));
	} else {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_3) * t_4);
	}
	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):
	t_2 = l / (k * math.pow(t_m, 1.5))
	t_3 = k / math.cos(k)
	t_4 = t_m * math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 0.55:
		tmp = t_2 * t_2
	elif k <= 1e+185:
		tmp = 2.0 / (-k / (-math.pow(l, 2.0) / (t_3 * t_4)))
	else:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_3) * t_4)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(k * (t_m ^ 1.5)))
	t_3 = Float64(k / cos(k))
	t_4 = Float64(t_m * (sin(k) ^ 2.0))
	tmp = 0.0
	if (k <= 0.55)
		tmp = Float64(t_2 * t_2);
	elseif (k <= 1e+185)
		tmp = Float64(2.0 / Float64(Float64(-k) / Float64(Float64(-(l ^ 2.0)) / Float64(t_3 * t_4))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * t_3) * t_4));
	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)
	t_2 = l / (k * (t_m ^ 1.5));
	t_3 = k / cos(k);
	t_4 = t_m * (sin(k) ^ 2.0);
	tmp = 0.0;
	if (k <= 0.55)
		tmp = t_2 * t_2;
	elseif (k <= 1e+185)
		tmp = 2.0 / (-k / (-(l ^ 2.0) / (t_3 * t_4)));
	else
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_3) * t_4);
	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_] := Block[{t$95$2 = N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 0.55], N[(t$95$2 * t$95$2), $MachinePrecision], If[LessEqual[k, 1e+185], N[(2.0 / N[((-k) / N[((-N[Power[l, 2.0], $MachinePrecision]) / N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.55000000000000004

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*r* 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/r* 55.9%

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

      5. associate-/l/ 55.9%

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

      6. associate-/r/ 55.8%

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

      7. associate-/r/ 55.8%

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

      8. associate-/l* 61.0%

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

      9. +-commutative 61.0%

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

      10. associate-+r+ 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 49.7%

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

      1. unpow2 49.7%

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

      2. add-sqr-sqrt 24.7%

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

      3. times-frac 28.4%

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

      4. *-commutative 28.4%

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

      5. sqrt-prod 28.4%

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

      6. sqrt-pow1 28.4%

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

      7. metadata-eval 28.4%

         \[\leadsto \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      8. unpow2 28.4%

         \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      9. sqrt-prod 9.9%

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

      10. add-sqr-sqrt 20.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      11. *-commutative 20.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\sqrt{\color{blue}{{t}^{3} \cdot {k}^{2}}}} \]

      12. sqrt-prod 21.5%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{{k}^{2}}}} \]

      13. sqrt-pow1 23.4%

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

      14. metadata-eval 23.4%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \]

      15. unpow2 23.4%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \]

      16. sqrt-prod 14.6%

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

      17. add-sqr-sqrt 36.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \]

   6. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot k}} \]

   if 0.55000000000000004 < k < 9.9999999999999998e184

   1. Initial program 54.6%

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

   2. Taylor expanded in t around 0 79.2%

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

      1. associate-/l* 75.4%

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

      2. associate-/r/ 71.6%

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

   4. Simplified 71.6%

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

      1. unpow2 71.6%

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

      2. times-frac 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. associate-*l* 83.2%

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

      2. frac-2neg 83.2%

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

      3. associate-*l/ 86.9%

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

   8. Applied egg-rr 86.9%

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

      1. associate-/l* 86.9%

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

      2. *-commutative 86.9%

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

   10. Simplified 86.9%

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

   if 9.9999999999999998e184 < k

   1. Initial program 42.3%

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

   2. Taylor expanded in t around 0 62.0%

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

      1. associate-/l* 61.5%

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

      2. associate-/r/ 62.0%

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

   4. Simplified 62.0%

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

      1. unpow2 62.0%

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

      2. times-frac 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. metadata-eval 70.7%

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

      3. unpow2 70.7%

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

      4. times-frac 89.0%

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

      5. metadata-eval 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

   10. Simplified 89.0%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.55:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{elif}\;k \leq 10^{+185}:\\ \;\;\;\;\frac{2}{\frac{-k}{\frac{-{\ell}^{2}}{\frac{k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternative 8: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{\cos k}\\ t_3 := \frac{\ell}{k \cdot {t_m}^{1.5}}\\ t_4 := t_m \cdot {\sin k}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.97:\\ \;\;\;\;t_3 \cdot t_3\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{+183}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_2 \cdot t_4\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot t_2\right) \cdot t_4}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ k (cos k)))
        (t_3 (/ l (* k (pow t_m 1.5))))
        (t_4 (* t_m (pow (sin k) 2.0))))
   (*
    t_s
    (if (<= k 0.97)
      (* t_3 t_3)
      (if (<= k 1.85e+183)
        (/ 2.0 (/ (* k (* t_2 t_4)) (pow l 2.0)))
        (/ 2.0 (* (* (* (/ k l) (/ 1.0 l)) t_2) t_4)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = k / cos(k);
	double t_3 = l / (k * pow(t_m, 1.5));
	double t_4 = t_m * pow(sin(k), 2.0);
	double tmp;
	if (k <= 0.97) {
		tmp = t_3 * t_3;
	} else if (k <= 1.85e+183) {
		tmp = 2.0 / ((k * (t_2 * t_4)) / pow(l, 2.0));
	} else {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_2) * t_4);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = k / cos(k)
    t_3 = l / (k * (t_m ** 1.5d0))
    t_4 = t_m * (sin(k) ** 2.0d0)
    if (k <= 0.97d0) then
        tmp = t_3 * t_3
    else if (k <= 1.85d+183) then
        tmp = 2.0d0 / ((k * (t_2 * t_4)) / (l ** 2.0d0))
    else
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * t_2) * t_4)
    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 t_2 = k / Math.cos(k);
	double t_3 = l / (k * Math.pow(t_m, 1.5));
	double t_4 = t_m * Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 0.97) {
		tmp = t_3 * t_3;
	} else if (k <= 1.85e+183) {
		tmp = 2.0 / ((k * (t_2 * t_4)) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_2) * t_4);
	}
	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):
	t_2 = k / math.cos(k)
	t_3 = l / (k * math.pow(t_m, 1.5))
	t_4 = t_m * math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 0.97:
		tmp = t_3 * t_3
	elif k <= 1.85e+183:
		tmp = 2.0 / ((k * (t_2 * t_4)) / math.pow(l, 2.0))
	else:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_2) * t_4)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / cos(k))
	t_3 = Float64(l / Float64(k * (t_m ^ 1.5)))
	t_4 = Float64(t_m * (sin(k) ^ 2.0))
	tmp = 0.0
	if (k <= 0.97)
		tmp = Float64(t_3 * t_3);
	elseif (k <= 1.85e+183)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_2 * t_4)) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * t_2) * t_4));
	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)
	t_2 = k / cos(k);
	t_3 = l / (k * (t_m ^ 1.5));
	t_4 = t_m * (sin(k) ^ 2.0);
	tmp = 0.0;
	if (k <= 0.97)
		tmp = t_3 * t_3;
	elseif (k <= 1.85e+183)
		tmp = 2.0 / ((k * (t_2 * t_4)) / (l ^ 2.0));
	else
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * t_2) * t_4);
	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_] := Block[{t$95$2 = N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 0.97], N[(t$95$3 * t$95$3), $MachinePrecision], If[LessEqual[k, 1.85e+183], N[(2.0 / N[(N[(k * N[(t$95$2 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.96999999999999997

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*r* 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/r* 55.9%

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

      5. associate-/l/ 55.9%

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

      6. associate-/r/ 55.8%

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

      7. associate-/r/ 55.8%

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

      8. associate-/l* 61.0%

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

      9. +-commutative 61.0%

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

      10. associate-+r+ 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 49.7%

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

      1. unpow2 49.7%

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

      2. add-sqr-sqrt 24.7%

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

      3. times-frac 28.4%

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

      4. *-commutative 28.4%

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

      5. sqrt-prod 28.4%

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

      6. sqrt-pow1 28.4%

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

      7. metadata-eval 28.4%

         \[\leadsto \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      8. unpow2 28.4%

         \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      9. sqrt-prod 9.9%

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

      10. add-sqr-sqrt 20.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      11. *-commutative 20.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\sqrt{\color{blue}{{t}^{3} \cdot {k}^{2}}}} \]

      12. sqrt-prod 21.5%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{{k}^{2}}}} \]

      13. sqrt-pow1 23.4%

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

      14. metadata-eval 23.4%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \]

      15. unpow2 23.4%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \]

      16. sqrt-prod 14.6%

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

      17. add-sqr-sqrt 36.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \]

   6. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot k}} \]

   if 0.96999999999999997 < k < 1.8500000000000001e183

   1. Initial program 54.6%

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

   2. Taylor expanded in t around 0 79.2%

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

      1. associate-/l* 75.4%

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

      2. associate-/r/ 71.6%

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

   4. Simplified 71.6%

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

      1. unpow2 71.6%

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

      2. times-frac 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. associate-*l* 83.2%

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

      2. associate-*l/ 86.9%

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

   8. Applied egg-rr 86.9%

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

   if 1.8500000000000001e183 < k

   1. Initial program 42.3%

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

   2. Taylor expanded in t around 0 62.0%

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

      1. associate-/l* 61.5%

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

      2. associate-/r/ 62.0%

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

   4. Simplified 62.0%

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

      1. unpow2 62.0%

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

      2. times-frac 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. metadata-eval 70.7%

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

      3. unpow2 70.7%

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

      4. times-frac 89.0%

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

      5. metadata-eval 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. *-commutative 89.0%

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

   10. Simplified 89.0%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.97:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{+183}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternative 9: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot {t_m}^{1.5}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.68:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* k (pow t_m 1.5)))))
   (*
    t_s
    (if (<= k 0.68)
      (* t_2 t_2)
      (/
       2.0
       (*
        (* (* (/ k l) (/ 1.0 l)) (/ k (cos k)))
        (* t_m (pow (sin k) 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * pow(t_m, 1.5));
	double tmp;
	if (k <= 0.68) {
		tmp = t_2 * t_2;
	} else {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * pow(sin(k), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (k * (t_m ** 1.5d0))
    if (k <= 0.68d0) then
        tmp = t_2 * t_2
    else
        tmp = 2.0d0 / ((((k / l) * (1.0d0 / l)) * (k / cos(k))) * (t_m * (sin(k) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * Math.pow(t_m, 1.5));
	double tmp;
	if (k <= 0.68) {
		tmp = t_2 * t_2;
	} else {
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / Math.cos(k))) * (t_m * Math.pow(Math.sin(k), 2.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l / (k * math.pow(t_m, 1.5))
	tmp = 0
	if k <= 0.68:
		tmp = t_2 * t_2
	else:
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / math.cos(k))) * (t_m * math.pow(math.sin(k), 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(k * (t_m ^ 1.5)))
	tmp = 0.0
	if (k <= 0.68)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(1.0 / l)) * Float64(k / cos(k))) * Float64(t_m * (sin(k) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l / (k * (t_m ^ 1.5));
	tmp = 0.0;
	if (k <= 0.68)
		tmp = t_2 * t_2;
	else
		tmp = 2.0 / ((((k / l) * (1.0 / l)) * (k / cos(k))) * (t_m * (sin(k) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 0.68], N[(t$95$2 * t$95$2), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.680000000000000049

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*r* 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/r* 55.9%

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

      5. associate-/l/ 55.9%

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

      6. associate-/r/ 55.8%

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

      7. associate-/r/ 55.8%

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

      8. associate-/l* 61.0%

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

      9. +-commutative 61.0%

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

      10. associate-+r+ 61.0%

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

      11. metadata-eval 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 49.7%

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

      1. unpow2 49.7%

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

      2. add-sqr-sqrt 24.7%

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

      3. times-frac 28.4%

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

      4. *-commutative 28.4%

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

      5. sqrt-prod 28.4%

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

      6. sqrt-pow1 28.4%

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

      7. metadata-eval 28.4%

         \[\leadsto \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      8. unpow2 28.4%

         \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      9. sqrt-prod 9.9%

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

      10. add-sqr-sqrt 20.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      11. *-commutative 20.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\sqrt{\color{blue}{{t}^{3} \cdot {k}^{2}}}} \]

      12. sqrt-prod 21.5%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{{k}^{2}}}} \]

      13. sqrt-pow1 23.4%

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

      14. metadata-eval 23.4%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \]

      15. unpow2 23.4%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \]

      16. sqrt-prod 14.6%

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

      17. add-sqr-sqrt 36.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \]

   6. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot k}} \]

   if 0.680000000000000049 < k

   1. Initial program 48.2%

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

   2. Taylor expanded in t around 0 70.2%

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

      1. associate-/l* 68.2%

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

      2. associate-/r/ 66.6%

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

   4. Simplified 66.6%

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

      1. unpow2 66.6%

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

      2. times-frac 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. metadata-eval 71.1%

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

      3. unpow2 71.1%

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

      4. times-frac 80.6%

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

      5. metadata-eval 80.6%

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

   8. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

   10. Simplified 80.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.68:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{1}{\ell}\right) \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternative 10: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot {t_m}^{1.5}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\cos k} \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(t_m \cdot {k}^{2}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* k (pow t_m 1.5)))))
   (*
    t_s
    (if (<= k 1.95e+30)
      (* t_2 t_2)
      (/ 2.0 (* (* (/ k (cos k)) (/ k (pow l 2.0))) (* t_m (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 t_2 = l / (k * pow(t_m, 1.5));
	double tmp;
	if (k <= 1.95e+30) {
		tmp = t_2 * t_2;
	} else {
		tmp = 2.0 / (((k / cos(k)) * (k / pow(l, 2.0))) * (t_m * pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (k * (t_m ** 1.5d0))
    if (k <= 1.95d+30) then
        tmp = t_2 * t_2
    else
        tmp = 2.0d0 / (((k / cos(k)) * (k / (l ** 2.0d0))) * (t_m * (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * Math.pow(t_m, 1.5));
	double tmp;
	if (k <= 1.95e+30) {
		tmp = t_2 * t_2;
	} else {
		tmp = 2.0 / (((k / Math.cos(k)) * (k / Math.pow(l, 2.0))) * (t_m * Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l / (k * math.pow(t_m, 1.5))
	tmp = 0
	if k <= 1.95e+30:
		tmp = t_2 * t_2
	else:
		tmp = 2.0 / (((k / math.cos(k)) * (k / math.pow(l, 2.0))) * (t_m * 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)
	t_2 = Float64(l / Float64(k * (t_m ^ 1.5)))
	tmp = 0.0
	if (k <= 1.95e+30)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / cos(k)) * Float64(k / (l ^ 2.0))) * Float64(t_m * (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l / (k * (t_m ^ 1.5));
	tmp = 0.0;
	if (k <= 1.95e+30)
		tmp = t_2 * t_2;
	else
		tmp = 2.0 / (((k / cos(k)) * (k / (l ^ 2.0))) * (t_m * (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.95e+30], N[(t$95$2 * t$95$2), $MachinePrecision], N[(2.0 / N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.95000000000000005e30

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r* 55.4%

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

      3. *-commutative 55.4%

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

      4. associate-/r* 55.4%

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

      5. associate-/l/ 55.4%

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

      6. associate-/r/ 55.3%

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

      7. associate-/r/ 55.3%

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

      8. associate-/l* 60.5%

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

      9. +-commutative 60.5%

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

      10. associate-+r+ 60.5%

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

      11. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in k around 0 49.8%

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

      1. unpow2 49.8%

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

      2. add-sqr-sqrt 24.9%

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

      3. times-frac 28.6%

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

      4. *-commutative 28.6%

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

      5. sqrt-prod 28.6%

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

      6. sqrt-pow1 28.6%

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

      7. metadata-eval 28.6%

         \[\leadsto \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      8. unpow2 28.6%

         \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      9. sqrt-prod 10.3%

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

      10. add-sqr-sqrt 20.9%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      11. *-commutative 20.9%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\sqrt{\color{blue}{{t}^{3} \cdot {k}^{2}}}} \]

      12. sqrt-prod 21.8%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{{k}^{2}}}} \]

      13. sqrt-pow1 23.7%

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

      14. metadata-eval 23.7%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \]

      15. unpow2 23.7%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \]

      16. sqrt-prod 15.0%

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

      17. add-sqr-sqrt 36.7%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \]

   6. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot k}} \]

   if 1.95000000000000005e30 < k

   1. Initial program 50.1%

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

   2. Taylor expanded in t around 0 71.0%

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

      1. associate-/l* 68.9%

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

      2. associate-/r/ 67.2%

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

   4. Simplified 67.2%

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

      1. unpow2 67.2%

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

      2. times-frac 71.9%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in k around 0 62.1%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\cos k} \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 11: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot {t_m}^{1.5}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.1 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t_m}}}\\ \mathbf{elif}\;t_m \leq 3.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t_m}^{1.5}}}{{t_m}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* k (pow t_m 1.5)))))
   (*
    t_s
    (if (<= t_m 4.1e-243)
      (/ 2.0 (/ (pow k 4.0) (/ (pow l 2.0) t_m)))
      (if (<= t_m 3.3e+52)
        (/ (/ (pow (/ l k) 2.0) (pow t_m 1.5)) (pow t_m 1.5))
        (* t_2 t_2))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * pow(t_m, 1.5));
	double tmp;
	if (t_m <= 4.1e-243) {
		tmp = 2.0 / (pow(k, 4.0) / (pow(l, 2.0) / t_m));
	} else if (t_m <= 3.3e+52) {
		tmp = (pow((l / k), 2.0) / pow(t_m, 1.5)) / pow(t_m, 1.5);
	} else {
		tmp = t_2 * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = l / (k * (t_m ** 1.5d0))
    if (t_m <= 4.1d-243) then
        tmp = 2.0d0 / ((k ** 4.0d0) / ((l ** 2.0d0) / t_m))
    else if (t_m <= 3.3d+52) then
        tmp = (((l / k) ** 2.0d0) / (t_m ** 1.5d0)) / (t_m ** 1.5d0)
    else
        tmp = t_2 * t_2
    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 t_2 = l / (k * Math.pow(t_m, 1.5));
	double tmp;
	if (t_m <= 4.1e-243) {
		tmp = 2.0 / (Math.pow(k, 4.0) / (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 3.3e+52) {
		tmp = (Math.pow((l / k), 2.0) / Math.pow(t_m, 1.5)) / Math.pow(t_m, 1.5);
	} else {
		tmp = t_2 * t_2;
	}
	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):
	t_2 = l / (k * math.pow(t_m, 1.5))
	tmp = 0
	if t_m <= 4.1e-243:
		tmp = 2.0 / (math.pow(k, 4.0) / (math.pow(l, 2.0) / t_m))
	elif t_m <= 3.3e+52:
		tmp = (math.pow((l / k), 2.0) / math.pow(t_m, 1.5)) / math.pow(t_m, 1.5)
	else:
		tmp = t_2 * t_2
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(k * (t_m ^ 1.5)))
	tmp = 0.0
	if (t_m <= 4.1e-243)
		tmp = Float64(2.0 / Float64((k ^ 4.0) / Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 3.3e+52)
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / (t_m ^ 1.5)) / (t_m ^ 1.5));
	else
		tmp = Float64(t_2 * t_2);
	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)
	t_2 = l / (k * (t_m ^ 1.5));
	tmp = 0.0;
	if (t_m <= 4.1e-243)
		tmp = 2.0 / ((k ^ 4.0) / ((l ^ 2.0) / t_m));
	elseif (t_m <= 3.3e+52)
		tmp = (((l / k) ^ 2.0) / (t_m ^ 1.5)) / (t_m ^ 1.5);
	else
		tmp = t_2 * t_2;
	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_] := Block[{t$95$2 = N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.1e-243], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+52], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * t$95$2), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.09999999999999981e-243

   1. Initial program 49.8%

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

   2. Taylor expanded in t around 0 64.2%

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

      1. associate-/l* 64.4%

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

      2. associate-/r/ 63.7%

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

   4. Simplified 63.7%

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

   5. Taylor expanded in k around 0 56.5%

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

      1. associate-/l* 57.9%

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

   7. Simplified 57.9%

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

   if 4.09999999999999981e-243 < t < 3.3e52

   1. Initial program 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*r* 62.2%

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

      3. *-commutative 62.2%

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

      4. associate-/r* 62.2%

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

      5. associate-/l/ 62.4%

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

      6. associate-/r/ 62.4%

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

      7. associate-/r/ 62.3%

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

      8. associate-/l* 69.9%

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

      9. +-commutative 69.9%

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

      10. associate-+r+ 69.9%

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

      11. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in k around 0 51.2%

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

      1. unpow2 51.2%

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

      2. *-commutative 51.2%

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

      3. times-frac 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. *-un-lft-identity 61.6%

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

      2. metadata-eval 61.6%

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

      3. unpow2 61.6%

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

      4. times-frac 65.9%

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

      5. metadata-eval 65.9%

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

   8. Applied egg-rr 65.9%

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

      1. associate-*l/ 65.8%

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

      2. metadata-eval 65.8%

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

      3. pow-sqr 65.7%

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

      4. associate-*r* 65.7%

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

      5. div-inv 65.7%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k}} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}} \]

      6. associate-/r* 71.9%

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5}}}{{t}^{1.5}}} \]

      7. pow1 71.9%

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

      8. pow1 71.9%

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

      9. pow-sqr 71.9%

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

      10. metadata-eval 71.9%

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

   10. Applied egg-rr 71.9%

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

   if 3.3e52 < t

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-*r* 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/r* 55.8%

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

      5. associate-/l/ 55.8%

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

      6. associate-/r/ 55.5%

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

      7. associate-/r/ 57.1%

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

      8. associate-/l* 58.6%

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

      9. +-commutative 58.6%

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

      10. associate-+r+ 58.6%

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

      11. metadata-eval 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in k around 0 47.1%

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

      1. unpow2 47.1%

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

      2. add-sqr-sqrt 47.1%

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

      3. times-frac 52.9%

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

      4. *-commutative 52.9%

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

      5. sqrt-prod 52.9%

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

      6. sqrt-pow1 52.9%

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

      7. metadata-eval 52.9%

         \[\leadsto \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      8. unpow2 52.9%

         \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      9. sqrt-prod 24.0%

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

      10. add-sqr-sqrt 51.2%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      11. *-commutative 51.2%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\sqrt{\color{blue}{{t}^{3} \cdot {k}^{2}}}} \]

      12. sqrt-prod 51.2%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{{k}^{2}}}} \]

      13. sqrt-pow1 57.6%

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

      14. metadata-eval 57.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \]

      15. unpow2 57.6%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \]

      16. sqrt-prod 37.0%

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

      17. add-sqr-sqrt 72.9%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \]

   6. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{1.5}}}{{t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}\\ \end{array} \]

Alternative 12: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot {t_m}^{1.5}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+30}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* k (pow t_m 1.5)))))
   (*
    t_s
    (if (<= k 5.2e+30)
      (* t_2 t_2)
      (/ 2.0 (/ (* t_m (pow k 4.0)) (pow l 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * pow(t_m, 1.5));
	double tmp;
	if (k <= 5.2e+30) {
		tmp = t_2 * t_2;
	} else {
		tmp = 2.0 / ((t_m * pow(k, 4.0)) / pow(l, 2.0));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (k * (t_m ** 1.5d0))
    if (k <= 5.2d+30) then
        tmp = t_2 * t_2
    else
        tmp = 2.0d0 / ((t_m * (k ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * Math.pow(t_m, 1.5));
	double tmp;
	if (k <= 5.2e+30) {
		tmp = t_2 * t_2;
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k, 4.0)) / Math.pow(l, 2.0));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l / (k * math.pow(t_m, 1.5))
	tmp = 0
	if k <= 5.2e+30:
		tmp = t_2 * t_2
	else:
		tmp = 2.0 / ((t_m * math.pow(k, 4.0)) / math.pow(l, 2.0))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(k * (t_m ^ 1.5)))
	tmp = 0.0
	if (k <= 5.2e+30)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k ^ 4.0)) / (l ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l / (k * (t_m ^ 1.5));
	tmp = 0.0;
	if (k <= 5.2e+30)
		tmp = t_2 * t_2;
	else
		tmp = 2.0 / ((t_m * (k ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 5.2e+30], N[(t$95$2 * t$95$2), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.19999999999999977e30

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r* 55.4%

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

      3. *-commutative 55.4%

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

      4. associate-/r* 55.4%

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

      5. associate-/l/ 55.4%

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

      6. associate-/r/ 55.3%

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

      7. associate-/r/ 55.3%

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

      8. associate-/l* 60.5%

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

      9. +-commutative 60.5%

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

      10. associate-+r+ 60.5%

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

      11. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in k around 0 49.8%

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

      1. unpow2 49.8%

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

      2. add-sqr-sqrt 24.9%

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

      3. times-frac 28.6%

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

      4. *-commutative 28.6%

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

      5. sqrt-prod 28.6%

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

      6. sqrt-pow1 28.6%

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

      7. metadata-eval 28.6%

         \[\leadsto \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      8. unpow2 28.6%

         \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      9. sqrt-prod 10.3%

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

      10. add-sqr-sqrt 20.9%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \cdot \frac{\ell}{\sqrt{{k}^{2} \cdot {t}^{3}}} \]

      11. *-commutative 20.9%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\sqrt{\color{blue}{{t}^{3} \cdot {k}^{2}}}} \]

      12. sqrt-prod 21.8%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{{k}^{2}}}} \]

      13. sqrt-pow1 23.7%

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

      14. metadata-eval 23.7%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{\color{blue}{1.5}} \cdot \sqrt{{k}^{2}}} \]

      15. unpow2 23.7%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \sqrt{\color{blue}{k \cdot k}}} \]

      16. sqrt-prod 15.0%

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

      17. add-sqr-sqrt 36.7%

          \[\leadsto \frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot \color{blue}{k}} \]

   6. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{\ell}{{t}^{1.5} \cdot k}} \]

   if 5.19999999999999977e30 < k

   1. Initial program 50.1%

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

   2. Taylor expanded in t around 0 71.0%

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

      1. associate-/l* 68.9%

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

      2. associate-/r/ 67.2%

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

   4. Simplified 67.2%

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

   5. Taylor expanded in k around 0 61.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]

Alternative 13: 61.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-198}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{k \cdot {t_m}^{3}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+120}:\\ \;\;\;\;\left(\frac{1}{t_m} \cdot \frac{\ell}{{t_m}^{2}}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.85e-198)
    (/ (* l (/ l k)) (* k (pow t_m 3.0)))
    (if (<= k 2.35e+120)
      (* (* (/ 1.0 t_m) (/ l (pow t_m 2.0))) (* (/ l k) (/ 1.0 k)))
      (/ 2.0 (/ (* t_m (pow k 4.0)) (pow l 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.85e-198) {
		tmp = (l * (l / k)) / (k * pow(t_m, 3.0));
	} else if (k <= 2.35e+120) {
		tmp = ((1.0 / t_m) * (l / pow(t_m, 2.0))) * ((l / k) * (1.0 / k));
	} else {
		tmp = 2.0 / ((t_m * pow(k, 4.0)) / pow(l, 2.0));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.85d-198) then
        tmp = (l * (l / k)) / (k * (t_m ** 3.0d0))
    else if (k <= 2.35d+120) then
        tmp = ((1.0d0 / t_m) * (l / (t_m ** 2.0d0))) * ((l / k) * (1.0d0 / k))
    else
        tmp = 2.0d0 / ((t_m * (k ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.85e-198) {
		tmp = (l * (l / k)) / (k * Math.pow(t_m, 3.0));
	} else if (k <= 2.35e+120) {
		tmp = ((1.0 / t_m) * (l / Math.pow(t_m, 2.0))) * ((l / k) * (1.0 / k));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k, 4.0)) / Math.pow(l, 2.0));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.85e-198:
		tmp = (l * (l / k)) / (k * math.pow(t_m, 3.0))
	elif k <= 2.35e+120:
		tmp = ((1.0 / t_m) * (l / math.pow(t_m, 2.0))) * ((l / k) * (1.0 / k))
	else:
		tmp = 2.0 / ((t_m * math.pow(k, 4.0)) / math.pow(l, 2.0))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.85e-198)
		tmp = Float64(Float64(l * Float64(l / k)) / Float64(k * (t_m ^ 3.0)));
	elseif (k <= 2.35e+120)
		tmp = Float64(Float64(Float64(1.0 / t_m) * Float64(l / (t_m ^ 2.0))) * Float64(Float64(l / k) * Float64(1.0 / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k ^ 4.0)) / (l ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.85e-198)
		tmp = (l * (l / k)) / (k * (t_m ^ 3.0));
	elseif (k <= 2.35e+120)
		tmp = ((1.0 / t_m) * (l / (t_m ^ 2.0))) * ((l / k) * (1.0 / k));
	else
		tmp = 2.0 / ((t_m * (k ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.85e-198], N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.35e+120], N[(N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.84999999999999986e-198

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r* 55.4%

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

      3. *-commutative 55.4%

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

      4. associate-/r* 55.4%

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

      5. associate-/l/ 55.4%

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

      6. associate-/r/ 55.9%

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

      7. associate-/r/ 56.5%

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

      8. associate-/l* 61.5%

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

      9. +-commutative 61.5%

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

      10. associate-+r+ 61.5%

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

      11. metadata-eval 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in k around 0 47.1%

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

      1. unpow2 47.1%

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

      2. *-commutative 47.1%

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

      3. times-frac 53.6%

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

   6. Applied egg-rr 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. metadata-eval 53.6%

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

      3. unpow2 53.6%

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

      4. times-frac 60.8%

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

      5. metadata-eval 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l/ 60.9%

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

      3. *-un-lft-identity 60.9%

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

      4. frac-times 60.3%

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

   10. Applied egg-rr 60.3%

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

   if 1.84999999999999986e-198 < k < 2.34999999999999997e120

   1. Initial program 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-*r* 57.4%

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

      3. *-commutative 57.4%

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

      4. associate-/r* 57.3%

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

      5. associate-/l/ 57.3%

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

      6. associate-/r/ 55.9%

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

      7. associate-/r/ 54.3%

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

      8. associate-/l* 58.8%

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

      9. +-commutative 58.8%

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

      10. associate-+r+ 58.8%

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

      11. metadata-eval 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in k around 0 56.4%

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

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 62.6%

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

   6. Applied egg-rr 62.6%

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

      1. *-un-lft-identity 62.6%

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

      2. metadata-eval 62.6%

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

      3. unpow2 62.6%

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

      4. times-frac 67.7%

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

      5. metadata-eval 67.7%

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

   8. Applied egg-rr 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. metadata-eval 67.7%

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

      3. cube-mult 67.7%

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

      4. times-frac 69.1%

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

      5. metadata-eval 69.1%

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

      6. pow1 69.1%

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

      7. metadata-eval 69.1%

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

      8. pow1 69.1%

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

      9. metadata-eval 69.1%

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

      10. pow-sqr 69.1%

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

      11. metadata-eval 69.1%

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

      12. metadata-eval 69.1%

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

   10. Applied egg-rr 69.1%

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

   if 2.34999999999999997e120 < k

   1. Initial program 44.1%

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

   2. Taylor expanded in t around 0 65.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 60.0%

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

   4. Simplified 60.0%

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

   5. Taylor expanded in k around 0 62.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-198}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{k \cdot {t}^{3}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+120}:\\ \;\;\;\;\left(\frac{1}{t} \cdot \frac{\ell}{{t}^{2}}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]

Alternative 14: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t_m}}{{t_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.4e+120)
    (/ (/ (pow (/ l k) 2.0) t_m) (pow t_m 2.0))
    (/ 2.0 (/ (* t_m (pow k 4.0)) (pow l 2.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e+120) {
		tmp = (pow((l / k), 2.0) / t_m) / pow(t_m, 2.0);
	} else {
		tmp = 2.0 / ((t_m * pow(k, 4.0)) / pow(l, 2.0));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.4d+120) then
        tmp = (((l / k) ** 2.0d0) / t_m) / (t_m ** 2.0d0)
    else
        tmp = 2.0d0 / ((t_m * (k ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e+120) {
		tmp = (Math.pow((l / k), 2.0) / t_m) / Math.pow(t_m, 2.0);
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k, 4.0)) / Math.pow(l, 2.0));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.4e+120:
		tmp = (math.pow((l / k), 2.0) / t_m) / math.pow(t_m, 2.0)
	else:
		tmp = 2.0 / ((t_m * math.pow(k, 4.0)) / math.pow(l, 2.0))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.4e+120)
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t_m) / (t_m ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k ^ 4.0)) / (l ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.4e+120)
		tmp = (((l / k) ^ 2.0) / t_m) / (t_m ^ 2.0);
	else
		tmp = 2.0 / ((t_m * (k ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.4e+120], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000001e120

   1. Initial program 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*r* 56.0%

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

      3. *-commutative 56.0%

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

      4. associate-/r* 55.9%

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

      5. associate-/l/ 56.0%

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

      6. associate-/r/ 55.9%

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

      7. associate-/r/ 55.8%

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

      8. associate-/l* 60.7%

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

      9. +-commutative 60.7%

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

      10. associate-+r+ 60.7%

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

      11. metadata-eval 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in k around 0 49.9%

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

      1. unpow2 49.9%

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

      2. *-commutative 49.9%

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

      3. times-frac 56.3%

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

   6. Applied egg-rr 56.3%

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

      1. *-un-lft-identity 56.3%

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

      2. metadata-eval 56.3%

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

      3. unpow2 56.3%

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

      4. times-frac 62.9%

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

      5. metadata-eval 62.9%

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

   8. Applied egg-rr 62.9%

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

      1. associate-*l/ 62.6%

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

      2. cube-mult 62.6%

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

      3. associate-*r* 64.0%

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

      4. div-inv 64.0%

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

      5. associate-/r* 67.9%

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

      6. pow1 67.9%

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

      7. pow1 67.9%

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

      8. pow-sqr 67.9%

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

      9. metadata-eval 67.9%

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

      10. pow1 67.9%

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

      11. pow1 67.9%

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

      12. pow-sqr 67.9%

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

      13. metadata-eval 67.9%

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

   10. Applied egg-rr 67.9%

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

   if 2.40000000000000001e120 < k

   1. Initial program 44.1%

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

   2. Taylor expanded in t around 0 65.4%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 60.0%

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

   4. Simplified 60.0%

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

   5. Taylor expanded in k around 0 62.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{t}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]

Alternative 15: 60.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-199}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{k \cdot {t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t_m} \cdot \frac{\ell}{{t_m}^{2}}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.3e-199)
    (/ (* l (/ l k)) (* k (pow t_m 3.0)))
    (* (* (/ 1.0 t_m) (/ l (pow t_m 2.0))) (* (/ l k) (/ 1.0 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 (k <= 3.3e-199) {
		tmp = (l * (l / k)) / (k * pow(t_m, 3.0));
	} else {
		tmp = ((1.0 / t_m) * (l / pow(t_m, 2.0))) * ((l / k) * (1.0 / 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 <= 3.3d-199) then
        tmp = (l * (l / k)) / (k * (t_m ** 3.0d0))
    else
        tmp = ((1.0d0 / t_m) * (l / (t_m ** 2.0d0))) * ((l / k) * (1.0d0 / 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 <= 3.3e-199) {
		tmp = (l * (l / k)) / (k * Math.pow(t_m, 3.0));
	} else {
		tmp = ((1.0 / t_m) * (l / Math.pow(t_m, 2.0))) * ((l / k) * (1.0 / 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 <= 3.3e-199:
		tmp = (l * (l / k)) / (k * math.pow(t_m, 3.0))
	else:
		tmp = ((1.0 / t_m) * (l / math.pow(t_m, 2.0))) * ((l / k) * (1.0 / 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 (k <= 3.3e-199)
		tmp = Float64(Float64(l * Float64(l / k)) / Float64(k * (t_m ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / t_m) * Float64(l / (t_m ^ 2.0))) * Float64(Float64(l / k) * Float64(1.0 / 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 <= 3.3e-199)
		tmp = (l * (l / k)) / (k * (t_m ^ 3.0));
	else
		tmp = ((1.0 / t_m) * (l / (t_m ^ 2.0))) * ((l / k) * (1.0 / 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, 3.3e-199], N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(l / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.3000000000000002e-199

   1. Initial program 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r* 55.4%

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

      3. *-commutative 55.4%

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

      4. associate-/r* 55.4%

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

      5. associate-/l/ 55.4%

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

      6. associate-/r/ 55.9%

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

      7. associate-/r/ 56.5%

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

      8. associate-/l* 61.5%

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

      9. +-commutative 61.5%

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

      10. associate-+r+ 61.5%

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

      11. metadata-eval 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in k around 0 47.1%

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

      1. unpow2 47.1%

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

      2. *-commutative 47.1%

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

      3. times-frac 53.6%

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

   6. Applied egg-rr 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. metadata-eval 53.6%

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

      3. unpow2 53.6%

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

      4. times-frac 60.8%

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

      5. metadata-eval 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*l/ 60.9%

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

      3. *-un-lft-identity 60.9%

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

      4. frac-times 60.3%

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

   10. Applied egg-rr 60.3%

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

   if 3.3000000000000002e-199 < k

   1. Initial program 52.9%

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

      1. *-commutative 52.9%

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

      2. associate-*r* 52.9%

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

      3. *-commutative 52.9%

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

      4. associate-/r* 52.9%

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

      5. associate-/l/ 52.9%

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

      6. associate-/r/ 51.9%

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

      7. associate-/r/ 50.9%

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

      8. associate-/l* 53.9%

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

      9. +-commutative 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 52.3%

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

      1. unpow2 52.3%

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

      2. *-commutative 52.3%

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

      3. times-frac 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. *-un-lft-identity 56.5%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}} \]

      2. metadata-eval 56.5%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{k}^{2}} \]

      3. unpow2 56.5%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{2}{2} \cdot \ell}{\color{blue}{k \cdot k}} \]

      4. times-frac 60.0%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{\frac{2}{2}}{k} \cdot \frac{\ell}{k}\right)} \]

      5. metadata-eval 60.0%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \left(\frac{\color{blue}{1}}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied egg-rr 60.0%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 60.0%

         \[\leadsto \frac{\color{blue}{1 \cdot \ell}}{{t}^{3}} \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      2. metadata-eval 60.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{t}^{3}} \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      3. cube-mult 60.0%

         \[\leadsto \frac{\frac{2}{2} \cdot \ell}{\color{blue}{t \cdot \left(t \cdot t\right)}} \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      4. times-frac 62.9%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{2}}{t} \cdot \frac{\ell}{t \cdot t}\right)} \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      5. metadata-eval 62.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{t} \cdot \frac{\ell}{t \cdot t}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      6. pow1 62.9%

         \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{\color{blue}{{t}^{1}} \cdot t}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      7. metadata-eval 62.9%

         \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{{t}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot t}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      8. pow1 62.9%

         \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{{t}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{t}^{1}}}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      9. metadata-eval 62.9%

         \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{{t}^{\left(\frac{2}{2}\right)} \cdot {t}^{\color{blue}{\left(\frac{2}{2}\right)}}}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      10. pow-sqr 62.9%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{\color{blue}{{t}^{\left(2 \cdot \frac{2}{2}\right)}}}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      11. metadata-eval 62.9%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{{t}^{\left(2 \cdot \color{blue}{1}\right)}}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

      12. metadata-eval 62.9%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell}{{t}^{\color{blue}{2}}}\right) \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]

   10. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{\ell}{{t}^{2}}\right)} \cdot \left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-199}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t} \cdot \frac{\ell}{{t}^{2}}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)\\ \end{array} \]

Alternative 16: 59.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{k \cdot {t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t_m}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-180)
    (/ (* l (/ l k)) (* k (pow t_m 3.0)))
    (* (/ l (pow t_m 3.0)) (/ (/ l k) k)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-180) {
		tmp = (l * (l / k)) / (k * pow(t_m, 3.0));
	} else {
		tmp = (l / pow(t_m, 3.0)) * ((l / k) / k);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d-180) then
        tmp = (l * (l / k)) / (k * (t_m ** 3.0d0))
    else
        tmp = (l / (t_m ** 3.0d0)) * ((l / k) / k)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-180) {
		tmp = (l * (l / k)) / (k * Math.pow(t_m, 3.0));
	} else {
		tmp = (l / Math.pow(t_m, 3.0)) * ((l / k) / k);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.5e-180:
		tmp = (l * (l / k)) / (k * math.pow(t_m, 3.0))
	else:
		tmp = (l / math.pow(t_m, 3.0)) * ((l / k) / k)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-180)
		tmp = Float64(Float64(l * Float64(l / k)) / Float64(k * (t_m ^ 3.0)));
	else
		tmp = Float64(Float64(l / (t_m ^ 3.0)) * Float64(Float64(l / k) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.5e-180)
		tmp = (l * (l / k)) / (k * (t_m ^ 3.0));
	else
		tmp = (l / (t_m ^ 3.0)) * ((l / k) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-180], N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.50000000000000013e-180

   1. Initial program 56.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* 56.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \tan k}} \]

      3. *-commutative 56.7%

         \[\leadsto \frac{2}{\color{blue}{\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]

      4. associate-/r* 56.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. associate-/l/ 56.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      6. associate-/r/ 57.2%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r/ 57.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/l* 62.6%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. +-commutative 62.6%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      10. associate-+r+ 62.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]

      11. metadata-eval 62.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 47.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 47.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 47.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 53.8%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   6. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 53.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}} \]

      2. metadata-eval 53.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{k}^{2}} \]

      3. unpow2 53.8%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{2}{2} \cdot \ell}{\color{blue}{k \cdot k}} \]

      4. times-frac 61.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{\frac{2}{2}}{k} \cdot \frac{\ell}{k}\right)} \]

      5. metadata-eval 61.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \left(\frac{\color{blue}{1}}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied egg-rr 61.9%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{{t}^{3}}} \]

      2. associate-*l/ 61.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

      3. *-un-lft-identity 61.9%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{k} \cdot \frac{\ell}{{t}^{3}} \]

      4. frac-times 61.4%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k \cdot {t}^{3}}} \]

   10. Applied egg-rr 61.4%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \ell}{k \cdot {t}^{3}}} \]

   if 6.50000000000000013e-180 < k

   1. Initial program 50.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 50.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* 50.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \tan k}} \]

      3. *-commutative 50.4%

         \[\leadsto \frac{2}{\color{blue}{\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]

      4. associate-/r* 50.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. associate-/l/ 50.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      6. associate-/r/ 49.4%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r/ 48.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/l* 51.4%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. +-commutative 51.4%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      10. associate-+r+ 51.4%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]

      11. metadata-eval 51.4%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 52.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 52.0%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 52.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 56.3%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   6. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 56.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}} \]

      2. metadata-eval 56.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{k}^{2}} \]

      3. unpow2 56.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{2}{2} \cdot \ell}{\color{blue}{k \cdot k}} \]

      4. times-frac 58.1%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{\frac{2}{2}}{k} \cdot \frac{\ell}{k}\right)} \]

      5. metadata-eval 58.1%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \left(\frac{\color{blue}{1}}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied egg-rr 58.1%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 58.2%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1 \cdot \frac{\ell}{k}}{k}} \]

      2. *-un-lft-identity 58.2%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{\ell}{k}}}{k} \]

   10. Applied egg-rr 58.2%

       \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k}}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 17: 60.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;\frac{\ell}{{t_m}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t_m}^{3}}}{k}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e+52)
    (* (/ l (pow t_m 3.0)) (/ (/ l k) k))
    (/ (* l (/ l (* k (pow t_m 3.0)))) 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 <= 1.85e+52) {
		tmp = (l / pow(t_m, 3.0)) * ((l / k) / k);
	} else {
		tmp = (l * (l / (k * pow(t_m, 3.0)))) / 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 (t_m <= 1.85d+52) then
        tmp = (l / (t_m ** 3.0d0)) * ((l / k) / k)
    else
        tmp = (l * (l / (k * (t_m ** 3.0d0)))) / 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 (t_m <= 1.85e+52) {
		tmp = (l / Math.pow(t_m, 3.0)) * ((l / k) / k);
	} else {
		tmp = (l * (l / (k * Math.pow(t_m, 3.0)))) / 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 t_m <= 1.85e+52:
		tmp = (l / math.pow(t_m, 3.0)) * ((l / k) / k)
	else:
		tmp = (l * (l / (k * math.pow(t_m, 3.0)))) / 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 <= 1.85e+52)
		tmp = Float64(Float64(l / (t_m ^ 3.0)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(l * Float64(l / Float64(k * (t_m ^ 3.0)))) / 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 (t_m <= 1.85e+52)
		tmp = (l / (t_m ^ 3.0)) * ((l / k) / k);
	else
		tmp = (l * (l / (k * (t_m ^ 3.0)))) / 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[t$95$m, 1.85e+52], N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e52

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* 54.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \tan k}} \]

      3. *-commutative 54.0%

         \[\leadsto \frac{2}{\color{blue}{\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]

      4. associate-/r* 53.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. associate-/l/ 54.0%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      6. associate-/r/ 54.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r/ 53.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/l* 58.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. +-commutative 58.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      10. associate-+r+ 58.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]

      11. metadata-eval 58.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 49.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 49.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 49.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 55.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   6. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}} \]

      2. metadata-eval 55.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{k}^{2}} \]

      3. unpow2 55.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{2}{2} \cdot \ell}{\color{blue}{k \cdot k}} \]

      4. times-frac 62.4%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{\frac{2}{2}}{k} \cdot \frac{\ell}{k}\right)} \]

      5. metadata-eval 62.4%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \left(\frac{\color{blue}{1}}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied egg-rr 62.4%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 62.4%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1 \cdot \frac{\ell}{k}}{k}} \]

      2. *-un-lft-identity 62.4%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{\ell}{k}}}{k} \]

   10. Applied egg-rr 62.4%

       \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

   if 1.85e52 < t

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*r* 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \tan k}} \]

      3. *-commutative 55.8%

         \[\leadsto \frac{2}{\color{blue}{\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]

      4. associate-/r* 55.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. associate-/l/ 55.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      6. associate-/r/ 55.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r/ 57.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-/l* 58.6%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. +-commutative 58.6%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      10. associate-+r+ 58.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]

      11. metadata-eval 58.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 47.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 47.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative 47.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac 50.9%

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   6. Applied egg-rr 50.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 50.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}} \]

      2. metadata-eval 50.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{k}^{2}} \]

      3. unpow2 50.9%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{2}{2} \cdot \ell}{\color{blue}{k \cdot k}} \]

      4. times-frac 54.4%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{\frac{2}{2}}{k} \cdot \frac{\ell}{k}\right)} \]

      5. metadata-eval 54.4%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \left(\frac{\color{blue}{1}}{k} \cdot \frac{\ell}{k}\right) \]

   8. Applied egg-rr 54.4%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 59.5%

         \[\leadsto \color{blue}{\left(\frac{\ell}{{t}^{3}} \cdot \frac{1}{k}\right) \cdot \frac{\ell}{k}} \]

      2. associate-*r/ 60.1%

         \[\leadsto \color{blue}{\frac{\left(\frac{\ell}{{t}^{3}} \cdot \frac{1}{k}\right) \cdot \ell}{k}} \]

      3. frac-times 61.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 1}{{t}^{3} \cdot k}} \cdot \ell}{k} \]

      4. *-commutative 61.6%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \ell}}{{t}^{3} \cdot k} \cdot \ell}{k} \]

      5. *-un-lft-identity 61.6%

         \[\leadsto \frac{\frac{\color{blue}{\ell}}{{t}^{3} \cdot k} \cdot \ell}{k} \]

      6. *-commutative 61.6%

         \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot {t}^{3}}} \cdot \ell}{k} \]

   10. Applied egg-rr 61.6%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot {t}^{3}} \cdot \ell}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}\\ \end{array} \]

Alternative 18: 57.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\frac{\ell}{{t_m}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (pow t_m 3.0)) (/ (/ l k) k))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / pow(t_m, 3.0)) * ((l / k) / k));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (t_m ** 3.0d0)) * ((l / k) / k))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / Math.pow(t_m, 3.0)) * ((l / k) / k));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / math.pow(t_m, 3.0)) * ((l / k) / k))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / (t_m ^ 3.0)) * Float64(Float64(l / k) / k)))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (t_m ^ 3.0)) * ((l / k) / k));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 54.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*r* 54.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right) \cdot \tan k}} \]

   3. *-commutative 54.4%

      \[\leadsto \frac{2}{\color{blue}{\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]

   4. associate-/r* 54.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   5. associate-/l/ 54.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   6. associate-/r/ 54.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-/r/ 54.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-/l* 58.5%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. +-commutative 58.5%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   10. associate-+r+ 58.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]

   11. metadata-eval 58.5%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

3. Simplified 58.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around 0 49.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. unpow2 49.2%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative 49.2%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac 54.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

6. Applied egg-rr 54.7%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 54.7%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}} \]

   2. metadata-eval 54.7%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{2}{2}} \cdot \ell}{{k}^{2}} \]

   3. unpow2 54.7%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{2}{2} \cdot \ell}{\color{blue}{k \cdot k}} \]

   4. times-frac 60.5%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{\frac{2}{2}}{k} \cdot \frac{\ell}{k}\right)} \]

   5. metadata-eval 60.5%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \left(\frac{\color{blue}{1}}{k} \cdot \frac{\ell}{k}\right) \]

8. Applied egg-rr 60.5%

   \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\ell}{k}\right)} \]
9. Step-by-step derivation

   1. associate-*l/ 60.5%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{1 \cdot \frac{\ell}{k}}{k}} \]

   2. *-un-lft-identity 60.5%

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\color{blue}{\frac{\ell}{k}}}{k} \]

10. Applied egg-rr 60.5%

    \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}} \]

11. Final simplification 60.5%

    \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k} \]

Reproduce

herbie shell --seed 2023336 
(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))))