Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 90.2%

Time: 23.9s

Alternatives: 16

Speedup: 2.0×

• 26.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% 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, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + t\_2\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(t\_2 + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 \cdot \sin k}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 6.4e-54)
      (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
      (if (<= t_m 9.8e+89)
        (/
         2.0
         (* (+ 2.0 t_2) (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
        (if (<= t_m 1.3e+200)
          (/
           2.0
           (*
            (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
            (+ 1.0 (+ t_2 1.0))))
          (/
           2.0
           (pow
            (*
             (/ t_m (pow (cbrt l) 2.0))
             (* (cbrt (tan k)) (cbrt (* 2.0 (sin k)))))
            3.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.4e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_2) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else if (t_m <= 1.3e+200) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (t_2 + 1.0)));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(tan(k)) * cbrt((2.0 * sin(k))))), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.4e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_2) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else if (t_m <= 1.3e+200) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (t_2 + 1.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.tan(k)) * Math.cbrt((2.0 * Math.sin(k))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 6.4e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 9.8e+89)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_2) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	elseif (t_m <= 1.3e+200)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(t_2 + 1.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(tan(k)) * cbrt(Float64(2.0 * sin(k))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.4e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.8e+89], N[(2.0 / N[(N[(2.0 + t$95$2), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+200], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 6.39999999999999997e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 6.39999999999999997e-54 < t < 9.79999999999999992e89

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/r* 85.5%

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

      3. associate-*l/ 88.4%

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

      4. associate-*r/ 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. associate-/l* 88.5%

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

      3. associate-*l/ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-/l* 88.4%

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

      6. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. distribute-lft1-in 88.4%

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

      4. associate-+l+ 88.4%

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

      5. metadata-eval 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-/r/ 93.9%

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

      8. associate-/l* 94.0%

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

   8. Simplified 94.0%

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

   if 9.79999999999999992e89 < t < 1.3000000000000001e200

   1. Initial program 70.8%

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

      1. associate-/r* 71.4%

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

      2. add-sqr-sqrt 71.4%

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

      3. pow2 71.4%

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

      4. associate-/r* 70.8%

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

      5. sqrt-div 70.8%

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

      6. sqrt-pow1 77.4%

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

      7. metadata-eval 77.4%

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

      8. sqrt-prod 52.8%

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

      9. add-sqr-sqrt 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.3000000000000001e200 < t

   1. Initial program 81.7%

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

      1. associate-/r* 88.3%

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

      2. +-commutative 88.3%

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

      3. associate-+r+ 88.3%

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

      4. metadata-eval 88.3%

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

      5. associate-*r* 88.3%

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

      6. add-cube-cbrt 88.3%

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

      7. pow3 88.3%

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

   4. Applied egg-rr 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*r* 62.7%

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

      3. cbrt-prod 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in t around inf 99.4%

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

      1. *-commutative 99.4%

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

   9. Simplified 99.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 \cdot \sin k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \sqrt[3]{\tan k}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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-54)
    (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (* (cbrt (* (sin k) (+ 2.0 (pow (/ k t_m) 2.0)))) (cbrt (tan k))))
      3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt((sin(k) * (2.0 + pow((k / t_m), 2.0)))) * cbrt(tan(k)))), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt((Math.sin(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * Math.cbrt(Math.tan(k)))), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.5e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(Float64(sin(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * cbrt(tan(k)))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.5e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.50000000000000008e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 2.50000000000000008e-54 < t

   1. Initial program 78.6%

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

      1. associate-/r* 82.6%

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

      2. +-commutative 82.6%

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

      3. associate-+r+ 82.6%

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

      4. metadata-eval 82.6%

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

      5. associate-*r* 82.5%

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

      6. add-cube-cbrt 82.5%

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

      7. pow3 82.5%

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

   4. Applied egg-rr 85.8%

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

      1. *-commutative 85.8%

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

      2. associate-*r* 85.8%

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

      3. cbrt-prod 96.1%

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

   6. Applied egg-rr 96.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\tan k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\left(2 + t\_2\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 3.4e-54)
      (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
      (if (<= t_m 2.2e+99)
        (/
         2.0
         (* (+ 2.0 t_2) (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
        (/
         2.0
         (*
          (+ 1.0 (+ t_2 1.0))
          (* (tan k) (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.4e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 2.2e+99) {
		tmp = 2.0 / ((2.0 + t_2) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0))));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.4e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 2.2e+99) {
		tmp = 2.0 / ((2.0 + t_2) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0))));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 3.4e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 2.2e+99)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_2) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.4e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e+99], N[(2.0 / N[(N[(2.0 + t$95$2), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.39999999999999987e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 3.39999999999999987e-54 < t < 2.19999999999999978e99

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/r* 85.5%

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

      3. associate-*l/ 88.4%

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

      4. associate-*r/ 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. associate-/l* 88.5%

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

      3. associate-*l/ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-/l* 88.4%

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

      6. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. distribute-lft1-in 88.4%

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

      4. associate-+l+ 88.4%

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

      5. metadata-eval 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-/r/ 93.9%

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

      8. associate-/l* 94.0%

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

   8. Simplified 94.0%

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

   if 2.19999999999999978e99 < t

   1. Initial program 76.1%

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

      1. associate-/r* 79.6%

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

      2. add-cube-cbrt 79.6%

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

      3. pow3 79.6%

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

      4. associate-/r* 76.1%

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

      5. cbrt-div 76.1%

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

      6. rem-cbrt-cube 79.5%

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

      7. cbrt-prod 93.8%

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

      8. pow2 93.8%

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

   4. Applied egg-rr 93.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {t\_m}^{0.75} \cdot \sqrt{k}\\ t_3 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + t\_3\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left(t\_3 + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{t\_2}}{t\_2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow t_m 0.75) (sqrt k))) (t_3 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 3.2e-54)
      (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
      (if (<= t_m 9.8e+89)
        (/
         2.0
         (* (+ 2.0 t_3) (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
        (if (<= t_m 4.5e+192)
          (/
           2.0
           (*
            (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
            (+ 1.0 (+ t_3 1.0))))
          (pow (/ (/ l t_2) t_2) 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 = pow(t_m, 0.75) * sqrt(k);
	double t_3 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.2e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_3) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else if (t_m <= 4.5e+192) {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (t_3 + 1.0)));
	} else {
		tmp = pow(((l / t_2) / t_2), 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) :: t_3
    real(8) :: tmp
    t_2 = (t_m ** 0.75d0) * sqrt(k)
    t_3 = (k / t_m) ** 2.0d0
    if (t_m <= 3.2d-54) then
        tmp = 2.0d0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ** 2.0d0))
    else if (t_m <= 9.8d+89) then
        tmp = 2.0d0 / ((2.0d0 + t_3) * ((tan(k) / l) * ((t_m ** 3.0d0) / (l / sin(k)))))
    else if (t_m <= 4.5d+192) then
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0))) * (1.0d0 + (t_3 + 1.0d0)))
    else
        tmp = ((l / t_2) / t_2) ** 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 = Math.pow(t_m, 0.75) * Math.sqrt(k);
	double t_3 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.2e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_3) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else if (t_m <= 4.5e+192) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (t_3 + 1.0)));
	} else {
		tmp = Math.pow(((l / t_2) / t_2), 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 = math.pow(t_m, 0.75) * math.sqrt(k)
	t_3 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 3.2e-54:
		tmp = 2.0 / ((t_m / math.cos(k)) * math.pow((k * (math.sin(k) / l)), 2.0))
	elif t_m <= 9.8e+89:
		tmp = 2.0 / ((2.0 + t_3) * ((math.tan(k) / l) * (math.pow(t_m, 3.0) / (l / math.sin(k)))))
	elif t_m <= 4.5e+192:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (t_3 + 1.0)))
	else:
		tmp = math.pow(((l / t_2) / t_2), 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((t_m ^ 0.75) * sqrt(k))
	t_3 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 3.2e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 9.8e+89)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_3) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	elseif (t_m <= 4.5e+192)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64(t_3 + 1.0))));
	else
		tmp = Float64(Float64(l / t_2) / t_2) ^ 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 = (t_m ^ 0.75) * sqrt(k);
	t_3 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 3.2e-54)
		tmp = 2.0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ^ 2.0));
	elseif (t_m <= 9.8e+89)
		tmp = 2.0 / ((2.0 + t_3) * ((tan(k) / l) * ((t_m ^ 3.0) / (l / sin(k)))));
	elseif (t_m <= 4.5e+192)
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))) * (1.0 + (t_3 + 1.0)));
	else
		tmp = ((l / t_2) / t_2) ^ 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[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.8e+89], N[(2.0 / N[(N[(2.0 + t$95$3), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+192], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.19999999999999998e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 3.19999999999999998e-54 < t < 9.79999999999999992e89

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/r* 85.5%

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

      3. associate-*l/ 88.4%

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

      4. associate-*r/ 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. associate-/l* 88.5%

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

      3. associate-*l/ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-/l* 88.4%

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

      6. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. distribute-lft1-in 88.4%

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

      4. associate-+l+ 88.4%

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

      5. metadata-eval 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-/r/ 93.9%

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

      8. associate-/l* 94.0%

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

   8. Simplified 94.0%

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

   if 9.79999999999999992e89 < t < 4.5e192

   1. Initial program 66.9%

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

      1. associate-/r* 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. pow2 67.6%

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

      4. associate-/r* 66.9%

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

      5. sqrt-div 66.9%

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

      6. sqrt-pow1 74.4%

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

      7. metadata-eval 74.4%

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

      8. sqrt-prod 46.5%

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

      9. add-sqr-sqrt 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 4.5e192 < t

   1. Initial program 83.7%

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

      1. associate-/r* 83.7%

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

      2. sqr-neg 83.7%

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

      3. associate-*l* 55.6%

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

      4. sqr-neg 55.6%

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

      5. associate-/r* 61.5%

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

      6. associate-+l+ 61.5%

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

      7. unpow2 61.5%

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

      8. times-frac 39.2%

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

      9. sqr-neg 39.2%

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

      10. times-frac 61.5%

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

      11. unpow2 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. expm1-log1p-u 55.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 89.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 89.6%

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

   9. Simplified 89.6%

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

       1. add-exp-log 84.1%

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

   11. Applied egg-rr 84.1%

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

       1. rem-exp-log 89.6%

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

       2. add-cube-cbrt 89.6%

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

       3. unpow2 89.6%

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

       4. add-sqr-sqrt 34.1%

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

       5. times-frac 34.1%

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

       6. *-commutative 34.1%

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

       7. sqrt-prod 34.1%

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

       8. sqrt-pow1 34.1%

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

       9. metadata-eval 34.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{\color{blue}{0.75}} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt{k \cdot {t}^{1.5}}}\right)}^{2} \]

       10. *-commutative 34.1%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt{\color{blue}{{t}^{1.5} \cdot k}}}\right)}^{2} \]

       11. sqrt-prod 34.1%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{\sqrt{{t}^{1.5}} \cdot \sqrt{k}}}\right)}^{2} \]

       12. sqrt-pow1 44.1%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}}\right)}^{2} \]

       13. metadata-eval 44.1%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{{t}^{\color{blue}{0.75}} \cdot \sqrt{k}}\right)}^{2} \]

   13. Applied egg-rr 44.1%

       \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}\right)}}^{2} \]
   14. Step-by-step derivation

       1. associate-*l/ 44.1%

          \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}}^{2} \]

       2. associate-*r/ 44.1%

          \[\leadsto {\left(\frac{\color{blue}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}^{2} \]

       3. unpow2 44.1%

          \[\leadsto {\left(\frac{\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot \sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}^{2} \]

       4. rem-3cbrt-lft 44.4%

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

   15. Simplified 44.4%

       \[\leadsto {\color{blue}{\left(\frac{\frac{\ell}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {t\_m}^{0.75} \cdot \sqrt{k}\\ t_3 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + t\_3\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.08 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_3 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{t\_2}}{t\_2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow t_m 0.75) (sqrt k))) (t_3 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 2.35e-54)
      (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
      (if (<= t_m 9e+89)
        (/
         2.0
         (* (+ 2.0 t_3) (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
        (if (<= t_m 1.08e+154)
          (/
           2.0
           (*
            (+ 1.0 (+ t_3 1.0))
            (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))
          (pow (/ (/ l t_2) t_2) 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 = pow(t_m, 0.75) * sqrt(k);
	double t_3 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.35e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 9e+89) {
		tmp = 2.0 / ((2.0 + t_3) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else if (t_m <= 1.08e+154) {
		tmp = 2.0 / ((1.0 + (t_3 + 1.0)) * (tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = pow(((l / t_2) / t_2), 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) :: t_3
    real(8) :: tmp
    t_2 = (t_m ** 0.75d0) * sqrt(k)
    t_3 = (k / t_m) ** 2.0d0
    if (t_m <= 2.35d-54) then
        tmp = 2.0d0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ** 2.0d0))
    else if (t_m <= 9d+89) then
        tmp = 2.0d0 / ((2.0d0 + t_3) * ((tan(k) / l) * ((t_m ** 3.0d0) / (l / sin(k)))))
    else if (t_m <= 1.08d+154) then
        tmp = 2.0d0 / ((1.0d0 + (t_3 + 1.0d0)) * (tan(k) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l)))))
    else
        tmp = ((l / t_2) / t_2) ** 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 = Math.pow(t_m, 0.75) * Math.sqrt(k);
	double t_3 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.35e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 9e+89) {
		tmp = 2.0 / ((2.0 + t_3) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else if (t_m <= 1.08e+154) {
		tmp = 2.0 / ((1.0 + (t_3 + 1.0)) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = Math.pow(((l / t_2) / t_2), 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 = math.pow(t_m, 0.75) * math.sqrt(k)
	t_3 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 2.35e-54:
		tmp = 2.0 / ((t_m / math.cos(k)) * math.pow((k * (math.sin(k) / l)), 2.0))
	elif t_m <= 9e+89:
		tmp = 2.0 / ((2.0 + t_3) * ((math.tan(k) / l) * (math.pow(t_m, 3.0) / (l / math.sin(k)))))
	elif t_m <= 1.08e+154:
		tmp = 2.0 / ((1.0 + (t_3 + 1.0)) * (math.tan(k) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l)))))
	else:
		tmp = math.pow(((l / t_2) / t_2), 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((t_m ^ 0.75) * sqrt(k))
	t_3 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 2.35e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 9e+89)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_3) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	elseif (t_m <= 1.08e+154)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_3 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))))));
	else
		tmp = Float64(Float64(l / t_2) / t_2) ^ 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 = (t_m ^ 0.75) * sqrt(k);
	t_3 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 2.35e-54)
		tmp = 2.0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ^ 2.0));
	elseif (t_m <= 9e+89)
		tmp = 2.0 / ((2.0 + t_3) * ((tan(k) / l) * ((t_m ^ 3.0) / (l / sin(k)))));
	elseif (t_m <= 1.08e+154)
		tmp = 2.0 / ((1.0 + (t_3 + 1.0)) * (tan(k) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l)))));
	else
		tmp = ((l / t_2) / t_2) ^ 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[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.35e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+89], N[(2.0 / N[(N[(2.0 + t$95$3), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.08e+154], N[(2.0 / N[(N[(1.0 + N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.35e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 2.35e-54 < t < 9e89

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/r* 85.5%

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

      3. associate-*l/ 88.4%

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

      4. associate-*r/ 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. associate-/l* 88.5%

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

      3. associate-*l/ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-/l* 88.4%

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

      6. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. distribute-lft1-in 88.4%

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

      4. associate-+l+ 88.4%

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

      5. metadata-eval 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-/r/ 93.9%

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

      8. associate-/l* 94.0%

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

   8. Simplified 94.0%

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

   if 9e89 < t < 1.08e154

   1. Initial program 69.6%

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

      1. unpow3 69.6%

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

      2. times-frac 99.6%

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

      3. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1.08e154 < t

   1. Initial program 80.4%

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

      1. associate-/r* 80.4%

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

      2. sqr-neg 80.4%

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

      3. associate-*l* 55.0%

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

      4. sqr-neg 55.0%

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

      5. associate-/r* 60.5%

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

      6. associate-+l+ 60.5%

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

      7. unpow2 60.5%

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

      8. times-frac 35.5%

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

      9. sqr-neg 35.5%

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

      10. times-frac 60.5%

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

      11. unpow2 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in k around 0 55.0%

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

      1. expm1-log1p-u 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 55.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 90.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 90.6%

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

   9. Simplified 90.6%

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

       1. add-exp-log 80.7%

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

   11. Applied egg-rr 80.7%

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

       1. rem-exp-log 90.6%

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

       2. add-cube-cbrt 90.5%

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

       3. unpow2 90.5%

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

       4. add-sqr-sqrt 35.5%

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

       5. times-frac 35.5%

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

       6. *-commutative 35.5%

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

       7. sqrt-prod 35.5%

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

       8. sqrt-pow1 35.5%

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

       9. metadata-eval 35.5%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{\color{blue}{0.75}} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt{k \cdot {t}^{1.5}}}\right)}^{2} \]

       10. *-commutative 35.5%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt{\color{blue}{{t}^{1.5} \cdot k}}}\right)}^{2} \]

       11. sqrt-prod 35.5%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{\sqrt{{t}^{1.5}} \cdot \sqrt{k}}}\right)}^{2} \]

       12. sqrt-pow1 44.5%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{\color{blue}{{t}^{\left(\frac{1.5}{2}\right)}} \cdot \sqrt{k}}\right)}^{2} \]

       13. metadata-eval 44.5%

           \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{{t}^{\color{blue}{0.75}} \cdot \sqrt{k}}\right)}^{2} \]

   13. Applied egg-rr 44.5%

       \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{t}^{0.75} \cdot \sqrt{k}} \cdot \frac{\sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}\right)}}^{2} \]
   14. Step-by-step derivation

       1. associate-*l/ 44.5%

          \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \frac{\sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}}^{2} \]

       2. associate-*r/ 44.5%

          \[\leadsto {\left(\frac{\color{blue}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}^{2} \]

       3. unpow2 44.5%

          \[\leadsto {\left(\frac{\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot \sqrt[3]{\ell}}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}^{2} \]

       4. rem-3cbrt-lft 44.8%

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

   15. Simplified 44.8%

       \[\leadsto {\color{blue}{\left(\frac{\frac{\ell}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{{t}^{0.75} \cdot \sqrt{k}}}{{t}^{0.75} \cdot \sqrt{k}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.86 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + t\_2\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.86e-54)
      (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
      (if (<= t_m 9.8e+89)
        (/
         2.0
         (* (+ 2.0 t_2) (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
        (if (<= t_m 1.15e+154)
          (/
           2.0
           (*
            (+ 1.0 (+ t_2 1.0))
            (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))
          (pow (/ (pow (cbrt l) 2.0) (* t_m (pow (cbrt k) 2.0))) 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.86e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_2) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else if (t_m <= 1.15e+154) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = pow((pow(cbrt(l), 2.0) / (t_m * pow(cbrt(k), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.86e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_2) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else if (t_m <= 1.15e+154) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t_m * Math.pow(Math.cbrt(k), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.86e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 9.8e+89)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_2) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	elseif (t_m <= 1.15e+154)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))))));
	else
		tmp = Float64((cbrt(l) ^ 2.0) / Float64(t_m * (cbrt(k) ^ 2.0))) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.86e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.8e+89], N[(2.0 / N[(N[(2.0 + t$95$2), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+154], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.8599999999999999e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 1.8599999999999999e-54 < t < 9.79999999999999992e89

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/r* 85.5%

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

      3. associate-*l/ 88.4%

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

      4. associate-*r/ 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. associate-/l* 88.5%

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

      3. associate-*l/ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-/l* 88.4%

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

      6. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. distribute-lft1-in 88.4%

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

      4. associate-+l+ 88.4%

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

      5. metadata-eval 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-/r/ 93.9%

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

      8. associate-/l* 94.0%

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

   8. Simplified 94.0%

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

   if 9.79999999999999992e89 < t < 1.15e154

   1. Initial program 69.6%

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

      1. unpow3 69.6%

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

      2. times-frac 99.6%

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

      3. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1.15e154 < t

   1. Initial program 80.4%

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

      1. associate-/r* 80.4%

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

      2. sqr-neg 80.4%

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

      3. associate-*l* 55.0%

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

      4. sqr-neg 55.0%

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

      5. associate-/r* 60.5%

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

      6. associate-+l+ 60.5%

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

      7. unpow2 60.5%

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

      8. times-frac 35.5%

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

      9. sqr-neg 35.5%

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

      10. times-frac 60.5%

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

      11. unpow2 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in k around 0 55.0%

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

      1. add-cube-cbrt 55.0%

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

      2. pow3 55.0%

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

      3. *-commutative 55.0%

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

      4. cbrt-prod 55.0%

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

      5. rem-cbrt-cube 55.4%

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

      6. unpow2 55.4%

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

      7. cbrt-prod 80.4%

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

      8. pow2 80.4%

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

   7. Applied egg-rr 80.4%

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

      1. add-cube-cbrt 80.4%

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

      2. pow2 80.4%

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

      3. cbrt-div 80.4%

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

      4. unpow2 80.4%

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

      5. cbrt-prod 80.4%

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

      6. unpow2 80.4%

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

      7. unpow3 80.4%

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

      8. add-cbrt-cube 80.4%

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

      9. cbrt-div 80.4%

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

      10. unpow2 80.4%

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

      11. cbrt-prod 85.8%

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

      12. unpow2 85.8%

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

      13. unpow3 85.8%

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

   9. Applied egg-rr 99.4%

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

       1. pow-plus 99.4%

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

       2. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.86 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + t\_2\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 2.35e-54)
      (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
      (if (<= t_m 9.8e+89)
        (/
         2.0
         (* (+ 2.0 t_2) (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
        (if (<= t_m 1.12e+154)
          (/
           2.0
           (*
            (+ 1.0 (+ t_2 1.0))
            (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))
          (pow (* l (pow (* (pow t_m 0.75) (sqrt k)) -2.0)) 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 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.35e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_2) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else if (t_m <= 1.12e+154) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = pow((l * pow((pow(t_m, 0.75) * sqrt(k)), -2.0)), 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 = (k / t_m) ** 2.0d0
    if (t_m <= 2.35d-54) then
        tmp = 2.0d0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ** 2.0d0))
    else if (t_m <= 9.8d+89) then
        tmp = 2.0d0 / ((2.0d0 + t_2) * ((tan(k) / l) * ((t_m ** 3.0d0) / (l / sin(k)))))
    else if (t_m <= 1.12d+154) then
        tmp = 2.0d0 / ((1.0d0 + (t_2 + 1.0d0)) * (tan(k) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l)))))
    else
        tmp = (l * (((t_m ** 0.75d0) * sqrt(k)) ** (-2.0d0))) ** 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 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.35e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((2.0 + t_2) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else if (t_m <= 1.12e+154) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = Math.pow((l * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -2.0)), 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 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 2.35e-54:
		tmp = 2.0 / ((t_m / math.cos(k)) * math.pow((k * (math.sin(k) / l)), 2.0))
	elif t_m <= 9.8e+89:
		tmp = 2.0 / ((2.0 + t_2) * ((math.tan(k) / l) * (math.pow(t_m, 3.0) / (l / math.sin(k)))))
	elif t_m <= 1.12e+154:
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (math.tan(k) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l)))))
	else:
		tmp = math.pow((l * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -2.0)), 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(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 2.35e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 9.8e+89)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_2) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	elseif (t_m <= 1.12e+154)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))))));
	else
		tmp = Float64(l * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 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 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 2.35e-54)
		tmp = 2.0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ^ 2.0));
	elseif (t_m <= 9.8e+89)
		tmp = 2.0 / ((2.0 + t_2) * ((tan(k) / l) * ((t_m ^ 3.0) / (l / sin(k)))));
	elseif (t_m <= 1.12e+154)
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l)))));
	else
		tmp = (l * (((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.35e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.8e+89], N[(2.0 / N[(N[(2.0 + t$95$2), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.12e+154], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.35e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 2.35e-54 < t < 9.79999999999999992e89

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-/r* 85.5%

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

      3. associate-*l/ 88.4%

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

      4. associate-*r/ 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. associate-/l* 88.5%

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

      3. associate-*l/ 88.5%

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

      4. +-commutative 88.5%

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

      5. associate-/l* 88.4%

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

      6. associate-*l/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

      2. *-rgt-identity 88.4%

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

      3. distribute-lft1-in 88.4%

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

      4. associate-+l+ 88.4%

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

      5. metadata-eval 88.4%

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

      6. +-commutative 88.4%

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

      7. associate-/r/ 93.9%

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

      8. associate-/l* 94.0%

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

   8. Simplified 94.0%

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

   if 9.79999999999999992e89 < t < 1.11999999999999994e154

   1. Initial program 69.6%

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

      1. unpow3 69.6%

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

      2. times-frac 99.6%

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

      3. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1.11999999999999994e154 < t

   1. Initial program 80.4%

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

      1. associate-/r* 80.4%

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

      2. sqr-neg 80.4%

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

      3. associate-*l* 55.0%

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

      4. sqr-neg 55.0%

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

      5. associate-/r* 60.5%

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

      6. associate-+l+ 60.5%

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

      7. unpow2 60.5%

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

      8. times-frac 35.5%

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

      9. sqr-neg 35.5%

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

      10. times-frac 60.5%

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

      11. unpow2 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in k around 0 55.0%

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

      1. expm1-log1p-u 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 55.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 90.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 90.6%

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

   9. Simplified 90.6%

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

       1. div-inv 90.6%

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

   11. Applied egg-rr 90.6%

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

       1. inv-pow 90.6%

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

       2. add-sqr-sqrt 35.6%

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

       3. unpow-prod-down 35.5%

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

       4. *-commutative 35.5%

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

       5. sqrt-prod 35.5%

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

       6. sqrt-pow1 35.5%

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

       7. metadata-eval 35.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 35.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 35.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 40.0%

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

       11. metadata-eval 40.0%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 40.0%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 40.2%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 40.2%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 40.2%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t\_m}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \end{array} \]

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 3.4e-54)
    (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) l)) 2.0)))
    (if (<= t_m 5.5e+102)
      (/
       2.0
       (*
        (+ 2.0 (pow (/ k t_m) 2.0))
        (* (/ (tan k) l) (/ (pow t_m 3.0) (/ l (sin k))))))
      (pow (* l (pow (* (pow t_m 0.75) (sqrt k)) -2.0)) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-54) {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / l)), 2.0));
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((2.0 + pow((k / t_m), 2.0)) * ((tan(k) / l) * (pow(t_m, 3.0) / (l / sin(k)))));
	} else {
		tmp = pow((l * pow((pow(t_m, 0.75) * sqrt(k)), -2.0)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.4d-54) then
        tmp = 2.0d0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ** 2.0d0))
    else if (t_m <= 5.5d+102) then
        tmp = 2.0d0 / ((2.0d0 + ((k / t_m) ** 2.0d0)) * ((tan(k) / l) * ((t_m ** 3.0d0) / (l / sin(k)))))
    else
        tmp = (l * (((t_m ** 0.75d0) * sqrt(k)) ** (-2.0d0))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-54) {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / l)), 2.0));
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((2.0 + Math.pow((k / t_m), 2.0)) * ((Math.tan(k) / l) * (Math.pow(t_m, 3.0) / (l / Math.sin(k)))));
	} else {
		tmp = Math.pow((l * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -2.0)), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.4e-54:
		tmp = 2.0 / ((t_m / math.cos(k)) * math.pow((k * (math.sin(k) / l)), 2.0))
	elif t_m <= 5.5e+102:
		tmp = 2.0 / ((2.0 + math.pow((k / t_m), 2.0)) * ((math.tan(k) / l) * (math.pow(t_m, 3.0) / (l / math.sin(k)))))
	else:
		tmp = math.pow((l * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -2.0)), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.4e-54)
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / l)) ^ 2.0)));
	elseif (t_m <= 5.5e+102)
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(Float64(tan(k) / l) * Float64((t_m ^ 3.0) / Float64(l / sin(k))))));
	else
		tmp = Float64(l * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.4e-54)
		tmp = 2.0 / ((t_m / cos(k)) * ((k * (sin(k) / l)) ^ 2.0));
	elseif (t_m <= 5.5e+102)
		tmp = 2.0 / ((2.0 + ((k / t_m) ^ 2.0)) * ((tan(k) / l) * ((t_m ^ 3.0) / (l / sin(k)))));
	else
		tmp = (l * (((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-54], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+102], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.39999999999999987e-54

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

      1. expm1-log1p-u 36.4%

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

      2. expm1-udef 30.4%

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

      3. *-commutative 30.4%

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

      4. unpow-prod-down 28.6%

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

      5. pow2 28.6%

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

      6. add-sqr-sqrt 28.7%

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

      7. associate-/l* 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. expm1-def 58.8%

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

       2. expm1-log1p 73.1%

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

       3. associate-/r/ 73.1%

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

       4. associate-*l/ 73.1%

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

       5. associate-*r/ 73.1%

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

   11. Simplified 73.1%

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

   if 3.39999999999999987e-54 < t < 5.49999999999999981e102

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-/r* 86.7%

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

      3. associate-*l/ 89.4%

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

      4. associate-*r/ 92.0%

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

   4. Applied egg-rr 92.0%

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

      1. distribute-lft-in 92.0%

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

      2. associate-/l* 89.4%

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

      3. associate-*l/ 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-/l* 89.4%

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

      6. associate-*l/ 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-commutative 89.4%

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

      2. *-rgt-identity 89.4%

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

      3. distribute-lft1-in 89.4%

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

      4. associate-+l+ 89.4%

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

      5. metadata-eval 89.4%

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

      6. +-commutative 89.4%

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

      7. associate-/r/ 94.4%

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

      8. associate-/l* 94.5%

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

   8. Simplified 94.5%

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

   if 5.49999999999999981e102 < t

   1. Initial program 73.7%

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

      1. associate-/r* 73.7%

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

      2. sqr-neg 73.7%

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

      3. associate-*l* 56.8%

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

      4. sqr-neg 56.8%

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

      5. associate-/r* 60.7%

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

      6. associate-+l+ 60.7%

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

      7. unpow2 60.7%

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

      8. times-frac 44.0%

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

      9. sqr-neg 44.0%

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

      10. times-frac 60.7%

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

      11. unpow2 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in k around 0 56.8%

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

      1. expm1-log1p-u 56.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 56.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 83.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 84.0%

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

   9. Simplified 84.0%

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

       1. div-inv 84.0%

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

   11. Applied egg-rr 84.0%

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

       1. inv-pow 84.0%

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

       2. add-sqr-sqrt 30.6%

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

       3. unpow-prod-down 30.5%

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

       4. *-commutative 30.5%

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

       5. sqrt-prod 30.5%

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

       6. sqrt-pow1 30.5%

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

       7. metadata-eval 30.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 30.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 30.5%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 33.5%

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

       11. metadata-eval 33.5%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 33.5%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 33.7%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 33.7%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 33.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\ell \cdot {\left({t\_m}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e-6)
    (pow (* l (pow (* (pow t_m 0.75) (sqrt k)) -2.0)) 2.0)
    (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) 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 <= 9.5e-6) {
		tmp = pow((l * pow((pow(t_m, 0.75) * sqrt(k)), -2.0)), 2.0);
	} else {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / 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 <= 9.5d-6) then
        tmp = (l * (((t_m ** 0.75d0) * sqrt(k)) ** (-2.0d0))) ** 2.0d0
    else
        tmp = 2.0d0 / ((t_m / cos(k)) * ((k * (sin(k) / 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 <= 9.5e-6) {
		tmp = Math.pow((l * Math.pow((Math.pow(t_m, 0.75) * Math.sqrt(k)), -2.0)), 2.0);
	} else {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / 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 <= 9.5e-6:
		tmp = math.pow((l * math.pow((math.pow(t_m, 0.75) * math.sqrt(k)), -2.0)), 2.0)
	else:
		tmp = 2.0 / ((t_m / math.cos(k)) * math.pow((k * (math.sin(k) / 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 <= 9.5e-6)
		tmp = Float64(l * (Float64((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / 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 <= 9.5e-6)
		tmp = (l * (((t_m ^ 0.75) * sqrt(k)) ^ -2.0)) ^ 2.0;
	else
		tmp = 2.0 / ((t_m / cos(k)) * ((k * (sin(k) / 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, 9.5e-6], N[Power[N[(l * N[Power[N[(N[Power[t$95$m, 0.75], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000005e-6

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

      2. sqr-neg 59.6%

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

      3. associate-*l* 54.3%

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

      4. sqr-neg 54.3%

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

      5. associate-/r* 59.4%

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

      6. associate-+l+ 59.4%

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

      7. unpow2 59.4%

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

      8. times-frac 44.7%

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

      9. sqr-neg 44.7%

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

      10. times-frac 59.4%

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

      11. unpow2 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. expm1-log1p-u 41.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 39.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 36.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 36.9%

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

   9. Simplified 36.9%

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

       1. div-inv 36.9%

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

   11. Applied egg-rr 36.9%

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

       1. inv-pow 36.9%

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

       2. add-sqr-sqrt 19.0%

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

       3. unpow-prod-down 19.0%

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

       4. *-commutative 19.0%

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

       5. sqrt-prod 15.2%

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

       6. sqrt-pow1 15.2%

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

       7. metadata-eval 15.2%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{k \cdot {t}^{1.5}}\right)}^{-1}\right)\right)}^{2} \]

       8. *-commutative 15.2%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{-1}\right)\right)}^{2} \]

       9. sqrt-prod 15.2%

          \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{{t}^{1.5}} \cdot \sqrt{k}\right)}}^{-1}\right)\right)}^{2} \]

       10. sqrt-pow1 15.7%

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

       11. metadata-eval 15.7%

           \[\leadsto {\left(\ell \cdot \left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{\color{blue}{0.75}} \cdot \sqrt{k}\right)}^{-1}\right)\right)}^{2} \]

   13. Applied egg-rr 15.7%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left({\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1} \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-1}\right)}\right)}^{2} \]
   14. Step-by-step derivation

       1. pow-sqr 15.7%

          \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\left(2 \cdot -1\right)}}\right)}^{2} \]

       2. metadata-eval 15.7%

          \[\leadsto {\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{\color{blue}{-2}}\right)}^{2} \]

   15. Simplified 15.7%

       \[\leadsto {\left(\ell \cdot \color{blue}{{\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}}\right)}^{2} \]

   if 9.5000000000000005e-6 < k

   1. Initial program 42.0%

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

      1. associate-/r* 49.3%

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

      2. +-commutative 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*r* 49.4%

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

      6. add-sqr-sqrt 32.3%

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

      7. pow2 32.3%

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

   4. Applied egg-rr 23.4%

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

      1. *-commutative 23.4%

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

   6. Simplified 23.4%

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

   7. Taylor expanded in k around inf 51.0%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 47.2%

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

      3. *-commutative 47.2%

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

      4. unpow-prod-down 44.8%

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

      5. pow2 44.8%

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

      6. add-sqr-sqrt 44.8%

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

      7. associate-/l* 44.8%

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

   9. Applied egg-rr 44.8%

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

       1. expm1-def 58.6%

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

       2. expm1-log1p 81.0%

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

       3. associate-/r/ 80.9%

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

       4. associate-*l/ 80.8%

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

       5. associate-*r/ 80.9%

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

   11. Simplified 80.9%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\ell \cdot {\left({t}^{0.75} \cdot \sqrt{k}\right)}^{-2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\ell}{{t\_m}^{1.5}} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-5)
    (pow (* (/ l (pow t_m 1.5)) (/ 1.0 k)) 2.0)
    (/ 2.0 (* (/ t_m (cos k)) (pow (* k (/ (sin k) 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 <= 1.25e-5) {
		tmp = pow(((l / pow(t_m, 1.5)) * (1.0 / k)), 2.0);
	} else {
		tmp = 2.0 / ((t_m / cos(k)) * pow((k * (sin(k) / 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 <= 1.25d-5) then
        tmp = ((l / (t_m ** 1.5d0)) * (1.0d0 / k)) ** 2.0d0
    else
        tmp = 2.0d0 / ((t_m / cos(k)) * ((k * (sin(k) / 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 <= 1.25e-5) {
		tmp = Math.pow(((l / Math.pow(t_m, 1.5)) * (1.0 / k)), 2.0);
	} else {
		tmp = 2.0 / ((t_m / Math.cos(k)) * Math.pow((k * (Math.sin(k) / 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 <= 1.25e-5:
		tmp = math.pow(((l / math.pow(t_m, 1.5)) * (1.0 / k)), 2.0)
	else:
		tmp = 2.0 / ((t_m / math.cos(k)) * math.pow((k * (math.sin(k) / 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 <= 1.25e-5)
		tmp = Float64(Float64(l / (t_m ^ 1.5)) * Float64(1.0 / k)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k)) * (Float64(k * Float64(sin(k) / 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 <= 1.25e-5)
		tmp = ((l / (t_m ^ 1.5)) * (1.0 / k)) ^ 2.0;
	else
		tmp = 2.0 / ((t_m / cos(k)) * ((k * (sin(k) / 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, 1.25e-5], N[Power[N[(N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000006e-5

   1. Initial program 59.6%

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

      1. associate-/r* 59.6%

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

      2. sqr-neg 59.6%

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

      3. associate-*l* 54.3%

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

      4. sqr-neg 54.3%

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

      5. associate-/r* 59.4%

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

      6. associate-+l+ 59.4%

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

      7. unpow2 59.4%

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

      8. times-frac 44.7%

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

      9. sqr-neg 44.7%

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

      10. times-frac 59.4%

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

      11. unpow2 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. expm1-log1p-u 41.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 39.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 35.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 36.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 36.9%

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

   9. Simplified 36.9%

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

       1. add-exp-log 22.5%

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

   11. Applied egg-rr 22.5%

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

       1. rem-exp-log 36.9%

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

       2. div-inv 36.9%

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

       3. associate-*r/ 36.9%

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

       4. *-commutative 36.9%

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

       5. times-frac 37.0%

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

   13. Applied egg-rr 37.0%

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

   if 1.25000000000000006e-5 < k

   1. Initial program 42.0%

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

      1. associate-/r* 49.3%

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

      2. +-commutative 49.3%

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

      3. associate-+r+ 49.3%

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

      4. metadata-eval 49.3%

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

      5. associate-*r* 49.4%

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

      6. add-sqr-sqrt 32.3%

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

      7. pow2 32.3%

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

   4. Applied egg-rr 23.4%

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

      1. *-commutative 23.4%

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

   6. Simplified 23.4%

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

   7. Taylor expanded in k around inf 51.0%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 47.2%

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

      3. *-commutative 47.2%

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

      4. unpow-prod-down 44.8%

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

      5. pow2 44.8%

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

      6. add-sqr-sqrt 44.8%

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

      7. associate-/l* 44.8%

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

   9. Applied egg-rr 44.8%

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

       1. expm1-def 58.6%

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

       2. expm1-log1p 81.0%

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

       3. associate-/r/ 80.9%

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

       4. associate-*l/ 80.8%

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

       5. associate-*r/ 80.9%

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

   11. Simplified 80.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{1.5}} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot {t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* k (pow t_m 1.5)) l)))
   (*
    t_s
    (if (<= t_m 5.3e-55)
      (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt t_m)) 2.0))
      (/ 1.0 (* 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 = (k * pow(t_m, 1.5)) / l;
	double tmp;
	if (t_m <= 5.3e-55) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = 1.0 / (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 = (k * (t_m ** 1.5d0)) / l
    if (t_m <= 5.3d-55) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 1.0d0 / (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 = (k * Math.pow(t_m, 1.5)) / l;
	double tmp;
	if (t_m <= 5.3e-55) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 1.0 / (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 = (k * math.pow(t_m, 1.5)) / l
	tmp = 0
	if t_m <= 5.3e-55:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 1.0 / (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(Float64(k * (t_m ^ 1.5)) / l)
	tmp = 0.0
	if (t_m <= 5.3e-55)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(1.0 / 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 = (k * (t_m ^ 1.5)) / l;
	tmp = 0.0;
	if (t_m <= 5.3e-55)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 1.0 / (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[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.3e-55], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.3000000000000003e-55

   1. Initial program 47.0%

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

      1. associate-/r* 53.2%

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

      2. +-commutative 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*r* 53.2%

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

      6. add-sqr-sqrt 18.6%

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

      7. pow2 18.6%

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

   4. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   6. Simplified 13.2%

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

   7. Taylor expanded in k around inf 36.8%

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

   8. Taylor expanded in k around 0 20.7%

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

   if 5.3000000000000003e-55 < t

   1. Initial program 78.6%

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

      1. associate-/r* 78.6%

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

      2. sqr-neg 78.6%

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

      3. associate-*l* 69.5%

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

      4. sqr-neg 69.5%

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

      5. associate-/r* 73.5%

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

      6. associate-+l+ 73.5%

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

      7. unpow2 73.5%

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

      8. times-frac 65.9%

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

      9. sqr-neg 65.9%

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

      10. times-frac 73.5%

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

      11. unpow2 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in k around 0 66.3%

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

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 63.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 83.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 84.0%

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

   9. Simplified 84.0%

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

       1. div-inv 84.0%

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

   11. Applied egg-rr 84.0%

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

       1. div-inv 84.0%

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

       2. pow2 84.0%

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

       3. clear-num 84.1%

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

       4. clear-num 84.0%

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

       5. frac-times 84.6%

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

       6. metadata-eval 84.6%

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

   13. Applied egg-rr 84.6%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k \cdot {t}^{1.5}}{\ell} \cdot \frac{k \cdot {t}^{1.5}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot {t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* k (pow t_m 1.5)) l)))
   (*
    t_s
    (if (<= t_m 1.9e-56)
      (/ 2.0 (pow (* (sqrt t_m) (/ (pow k 2.0) l)) 2.0))
      (/ 1.0 (* 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 = (k * pow(t_m, 1.5)) / l;
	double tmp;
	if (t_m <= 1.9e-56) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k, 2.0) / l)), 2.0);
	} else {
		tmp = 1.0 / (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 = (k * (t_m ** 1.5d0)) / l
    if (t_m <= 1.9d-56) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k ** 2.0d0) / l)) ** 2.0d0)
    else
        tmp = 1.0d0 / (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 = (k * Math.pow(t_m, 1.5)) / l;
	double tmp;
	if (t_m <= 1.9e-56) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k, 2.0) / l)), 2.0);
	} else {
		tmp = 1.0 / (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 = (k * math.pow(t_m, 1.5)) / l
	tmp = 0
	if t_m <= 1.9e-56:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k, 2.0) / l)), 2.0)
	else:
		tmp = 1.0 / (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(Float64(k * (t_m ^ 1.5)) / l)
	tmp = 0.0
	if (t_m <= 1.9e-56)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k ^ 2.0) / l)) ^ 2.0));
	else
		tmp = Float64(1.0 / 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 = (k * (t_m ^ 1.5)) / l;
	tmp = 0.0;
	if (t_m <= 1.9e-56)
		tmp = 2.0 / ((sqrt(t_m) * ((k ^ 2.0) / l)) ^ 2.0);
	else
		tmp = 1.0 / (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[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-56], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9000000000000001e-56

   1. Initial program 46.1%

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

      1. associate-/r* 52.5%

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

      2. +-commutative 52.5%

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

      3. associate-+r+ 52.5%

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

      4. metadata-eval 52.5%

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

      5. associate-*r* 52.5%

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

      6. add-sqr-sqrt 18.4%

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

      7. pow2 18.4%

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

   4. Applied egg-rr 12.9%

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

      1. *-commutative 12.9%

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

   6. Simplified 12.9%

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

   7. Taylor expanded in k around inf 36.8%

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

   8. Taylor expanded in k around 0 17.3%

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

   if 1.9000000000000001e-56 < t

   1. Initial program 79.5%

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

      1. associate-/r* 79.5%

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

      2. sqr-neg 79.5%

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

      3. associate-*l* 70.8%

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

      4. sqr-neg 70.8%

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

      5. associate-/r* 74.6%

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

      6. associate-+l+ 74.6%

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

      7. unpow2 74.6%

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

      8. times-frac 67.4%

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

      9. sqr-neg 67.4%

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

      10. times-frac 74.6%

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

      11. unpow2 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in k around 0 66.3%

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

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 63.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 83.3%

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

   9. Simplified 83.3%

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

       1. div-inv 83.3%

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

   11. Applied egg-rr 83.3%

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

       1. div-inv 83.3%

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

       2. pow2 83.3%

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

       3. clear-num 83.3%

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

       4. clear-num 83.3%

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

       5. frac-times 83.8%

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

       6. metadata-eval 83.8%

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

   13. Applied egg-rr 83.8%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k \cdot {t}^{1.5}}{\ell} \cdot \frac{k \cdot {t}^{1.5}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot {t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* k (pow t_m 1.5)) l)))
   (*
    t_s
    (if (<= t_m 1.55e-56)
      (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k 4.0)))
      (/ 1.0 (* 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 = (k * pow(t_m, 1.5)) / l;
	double tmp;
	if (t_m <= 1.55e-56) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0));
	} else {
		tmp = 1.0 / (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 = (k * (t_m ** 1.5d0)) / l
    if (t_m <= 1.55d-56) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0))
    else
        tmp = 1.0d0 / (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 = (k * Math.pow(t_m, 1.5)) / l;
	double tmp;
	if (t_m <= 1.55e-56) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0));
	} else {
		tmp = 1.0 / (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 = (k * math.pow(t_m, 1.5)) / l
	tmp = 0
	if t_m <= 1.55e-56:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0))
	else:
		tmp = 1.0 / (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(Float64(k * (t_m ^ 1.5)) / l)
	tmp = 0.0
	if (t_m <= 1.55e-56)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0)));
	else
		tmp = Float64(1.0 / 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 = (k * (t_m ^ 1.5)) / l;
	tmp = 0.0;
	if (t_m <= 1.55e-56)
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0));
	else
		tmp = 1.0 / (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[(N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-56], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.54999999999999994e-56

   1. Initial program 46.1%

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

      1. associate-/r* 46.1%

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

      2. sqr-neg 46.1%

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

      3. associate-*l* 43.9%

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

      4. sqr-neg 43.9%

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

      5. associate-/r* 50.0%

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

      6. associate-+l+ 50.0%

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

      7. unpow2 50.0%

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

      8. times-frac 33.4%

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

      9. sqr-neg 33.4%

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

      10. times-frac 50.0%

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

      11. unpow2 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 62.2%

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

   6. Taylor expanded in k around 0 45.3%

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

      1. *-commutative 45.3%

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

      2. associate-/r* 47.6%

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

   8. Simplified 47.6%

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

   if 1.54999999999999994e-56 < t

   1. Initial program 79.5%

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

      1. associate-/r* 79.5%

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

      2. sqr-neg 79.5%

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

      3. associate-*l* 70.8%

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

      4. sqr-neg 70.8%

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

      5. associate-/r* 74.6%

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

      6. associate-+l+ 74.6%

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

      7. unpow2 74.6%

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

      8. times-frac 67.4%

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

      9. sqr-neg 67.4%

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

      10. times-frac 74.6%

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

      11. unpow2 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in k around 0 66.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 63.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 83.3%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 83.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. div-inv 83.3%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

   11. Applied egg-rr 83.3%

       \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. div-inv 83.3%

          \[\leadsto {\color{blue}{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}}^{2} \]

       2. pow2 83.3%

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

       3. clear-num 83.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot {t}^{1.5}}{\ell}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

       4. clear-num 83.3%

          \[\leadsto \frac{1}{\frac{k \cdot {t}^{1.5}}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k \cdot {t}^{1.5}}{\ell}}} \]

       5. frac-times 83.8%

          \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{k \cdot {t}^{1.5}}{\ell} \cdot \frac{k \cdot {t}^{1.5}}{\ell}}} \]

       6. metadata-eval 83.8%

          \[\leadsto \frac{\color{blue}{1}}{\frac{k \cdot {t}^{1.5}}{\ell} \cdot \frac{k \cdot {t}^{1.5}}{\ell}} \]

   13. Applied egg-rr 83.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot {t}^{1.5}}{\ell} \cdot \frac{k \cdot {t}^{1.5}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k \cdot {t}^{1.5}}{\ell} \cdot \frac{k \cdot {t}^{1.5}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t\_m}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.45e-56)
    (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k 4.0)))
    (pow (/ l (* k (pow t_m 1.5))) 2.0))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.45e-56) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0));
	} else {
		tmp = pow((l / (k * pow(t_m, 1.5))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.45d-56) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0))
    else
        tmp = (l / (k * (t_m ** 1.5d0))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.45e-56) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0));
	} else {
		tmp = Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.45e-56:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0))
	else:
		tmp = math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.45e-56)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0)));
	else
		tmp = Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.45e-56)
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0));
	else
		tmp = (l / (k * (t_m ^ 1.5))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-56], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999996e-56

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 46.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 46.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 43.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 43.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 50.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 50.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 33.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 33.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 50.0%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 50.0%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 62.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 45.3%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      2. associate-/r* 47.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 47.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   if 1.44999999999999996e-56 < t

   1. Initial program 79.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 79.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 79.5%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 70.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 70.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 74.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 74.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 74.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 67.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 67.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 74.6%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 74.6%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 66.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 63.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 83.3%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 83.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{{t\_m}^{1.5}} \cdot \frac{1}{k}\right)}^{2}\\ \end{array} \end{array} \]

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 1.7e-56)
    (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k 4.0)))
    (pow (* (/ l (pow t_m 1.5)) (/ 1.0 k)) 2.0))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-56) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k, 4.0));
	} else {
		tmp = pow(((l / pow(t_m, 1.5)) * (1.0 / k)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d-56) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k ** 4.0d0))
    else
        tmp = ((l / (t_m ** 1.5d0)) * (1.0d0 / k)) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-56) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0));
	} else {
		tmp = Math.pow(((l / Math.pow(t_m, 1.5)) * (1.0 / k)), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e-56:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k, 4.0))
	else:
		tmp = math.pow(((l / math.pow(t_m, 1.5)) * (1.0 / k)), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-56)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l / (t_m ^ 1.5)) * Float64(1.0 / k)) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e-56)
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k ^ 4.0));
	else
		tmp = ((l / (t_m ^ 1.5)) * (1.0 / k)) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-56], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999991e-56

   1. Initial program 46.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 46.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 46.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 43.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 43.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 50.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 50.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 50.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 33.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 33.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 50.0%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 50.0%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 62.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 45.3%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      2. associate-/r* 47.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 47.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   if 1.69999999999999991e-56 < t

   1. Initial program 79.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 79.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 79.5%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 70.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 70.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 74.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 74.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 74.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 67.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 67.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 74.6%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 74.6%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 66.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

      2. expm1-udef 63.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

   7. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

      2. expm1-log1p 83.3%

         \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   9. Simplified 83.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   10. Step-by-step derivation

       1. add-exp-log 60.8%

          \[\leadsto {\color{blue}{\left(e^{\log \left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}\right)}}^{2} \]

   11. Applied egg-rr 60.8%

       \[\leadsto {\color{blue}{\left(e^{\log \left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}\right)}}^{2} \]
   12. Step-by-step derivation

       1. rem-exp-log 83.3%

          \[\leadsto {\color{blue}{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}}^{2} \]

       2. div-inv 83.3%

          \[\leadsto {\color{blue}{\left(\ell \cdot \frac{1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

       3. associate-*r/ 83.3%

          \[\leadsto {\color{blue}{\left(\frac{\ell \cdot 1}{k \cdot {t}^{1.5}}\right)}}^{2} \]

       4. *-commutative 83.3%

          \[\leadsto {\left(\frac{\ell \cdot 1}{\color{blue}{{t}^{1.5} \cdot k}}\right)}^{2} \]

       5. times-frac 83.3%

          \[\leadsto {\color{blue}{\left(\frac{\ell}{{t}^{1.5}} \cdot \frac{1}{k}\right)}}^{2} \]

   13. Applied egg-rr 83.3%

       \[\leadsto {\color{blue}{\left(\frac{\ell}{{t}^{1.5}} \cdot \frac{1}{k}\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{1.5}} \cdot \frac{1}{k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\frac{\ell}{k \cdot {t\_m}^{1.5}}\right)}^{2} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (pow (/ l (* k (pow t_m 1.5))) 2.0)))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * pow((l / (k * pow(t_m, 1.5))), 2.0);
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (k * (t_m ** 1.5d0))) ** 2.0d0)
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0);
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (k * (t_m ^ 1.5))) ^ 2.0);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 55.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. sqr-neg 55.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. associate-*l* 51.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. sqr-neg 51.2%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. associate-/r* 56.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. associate-+l+ 56.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

   7. unpow2 56.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

   8. times-frac 42.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

   9. sqr-neg 42.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

   10. times-frac 56.6%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

   11. unpow2 56.6%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 51.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 40.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

   2. expm1-udef 39.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

7. Applied egg-rr 32.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 33.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

   2. expm1-log1p 33.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

9. Simplified 33.3%

   \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

10. Final simplification 33.3%

    \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))