Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.6% → 79.8%

Time: 17.3s

Alternatives: 14

Speedup: 3.6×

• Could not converge on a ground truth

  1. t = 3.9725223424195e-140
  2. l = 2.3170149636677226e-71
  3. k = 1.6071684738669234e+78

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% 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: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k\_m}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\ell \cdot \sqrt{\frac{2}{\tan k\_m}}}\\ \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-20}:\\ \;\;\;\;\frac{{\left(\frac{t\_2 \cdot t\_2}{t \cdot \sqrt[3]{\sin k\_m}}\right)}^{3}}{t\_1}\\ \mathbf{elif}\;k\_m \leq 3.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\ell}}{\sqrt[3]{\ell}}\right) \cdot \left(t\_1 \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k_m t) 2.0)))
        (t_2 (cbrt (* l (sqrt (/ 2.0 (tan k_m)))))))
   (if (<= k_m 1.05e-20)
     (/ (pow (/ (* t_2 t_2) (* t (cbrt (sin k_m)))) 3.0) t_1)
     (if (<= k_m 3.4e+154)
       (/
        2.0
        (/
         (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))
         (* (pow l 2.0) (cos k_m))))
       (/
        2.0
        (*
         (* (pow (/ t (cbrt l)) 2.0) (/ (/ t l) (cbrt l)))
         (* t_1 (* (tan k_m) (sin k_m)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 + pow((k_m / t), 2.0);
	double t_2 = cbrt((l * sqrt((2.0 / tan(k_m)))));
	double tmp;
	if (k_m <= 1.05e-20) {
		tmp = pow(((t_2 * t_2) / (t * cbrt(sin(k_m)))), 3.0) / t_1;
	} else if (k_m <= 3.4e+154) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))) / (pow(l, 2.0) * cos(k_m)));
	} else {
		tmp = 2.0 / ((pow((t / cbrt(l)), 2.0) * ((t / l) / cbrt(l))) * (t_1 * (tan(k_m) * sin(k_m))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 2.0 + Math.pow((k_m / t), 2.0);
	double t_2 = Math.cbrt((l * Math.sqrt((2.0 / Math.tan(k_m)))));
	double tmp;
	if (k_m <= 1.05e-20) {
		tmp = Math.pow(((t_2 * t_2) / (t * Math.cbrt(Math.sin(k_m)))), 3.0) / t_1;
	} else if (k_m <= 3.4e+154) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	} else {
		tmp = 2.0 / ((Math.pow((t / Math.cbrt(l)), 2.0) * ((t / l) / Math.cbrt(l))) * (t_1 * (Math.tan(k_m) * Math.sin(k_m))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 + (Float64(k_m / t) ^ 2.0))
	t_2 = cbrt(Float64(l * sqrt(Float64(2.0 / tan(k_m)))))
	tmp = 0.0
	if (k_m <= 1.05e-20)
		tmp = Float64((Float64(Float64(t_2 * t_2) / Float64(t * cbrt(sin(k_m)))) ^ 3.0) / t_1);
	elseif (k_m <= 3.4e+154)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(t / cbrt(l)) ^ 2.0) * Float64(Float64(t / l) / cbrt(l))) * Float64(t_1 * Float64(tan(k_m) * sin(k_m)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l * N[Sqrt[N[(2.0 / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[k$95$m, 1.05e-20], N[(N[Power[N[(N[(t$95$2 * t$95$2), $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[k$95$m, 3.4e+154], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0499999999999999e-20

   1. Initial program 62.1%

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

   3. Simplified 61.6%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

      3. associate-*l/ 62.3%

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

      4. cbrt-div 62.3%

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

      5. pow2 62.3%

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

      6. cbrt-prod 62.3%

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

      7. rem-cbrt-cube 72.9%

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

   6. Applied egg-rr 72.9%

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

      1. pow1/3 43.5%

         \[\leadsto \frac{{\left(\frac{\color{blue}{{\left(\frac{2}{\tan k} \cdot {\ell}^{2}\right)}^{0.3333333333333333}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. add-sqr-sqrt 43.5%

         \[\leadsto \frac{{\left(\frac{{\color{blue}{\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}} \cdot \sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. unpow-prod-down 43.5%

         \[\leadsto \frac{{\left(\frac{\color{blue}{{\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. *-commutative 43.5%

         \[\leadsto \frac{{\left(\frac{{\left(\sqrt{\color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k}}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. sqrt-prod 30.8%

         \[\leadsto \frac{{\left(\frac{{\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{\frac{2}{\tan k}}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      6. sqrt-pow1 15.0%

         \[\leadsto \frac{{\left(\frac{{\left(\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. metadata-eval 15.0%

         \[\leadsto \frac{{\left(\frac{{\left({\ell}^{\color{blue}{1}} \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. pow1 15.0%

         \[\leadsto \frac{{\left(\frac{{\left(\color{blue}{\ell} \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\frac{2}{\tan k} \cdot {\ell}^{2}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      9. *-commutative 15.0%

         \[\leadsto \frac{{\left(\frac{{\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\color{blue}{{\ell}^{2} \cdot \frac{2}{\tan k}}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      10. sqrt-prod 15.9%

          \[\leadsto \frac{{\left(\frac{{\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\color{blue}{\left(\sqrt{{\ell}^{2}} \cdot \sqrt{\frac{2}{\tan k}}\right)}}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      11. sqrt-pow1 18.5%

          \[\leadsto \frac{{\left(\frac{{\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      12. metadata-eval 18.5%

          \[\leadsto \frac{{\left(\frac{{\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left({\ell}^{\color{blue}{1}} \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      13. pow1 18.5%

          \[\leadsto \frac{{\left(\frac{{\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\color{blue}{\ell} \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Applied egg-rr 18.5%

      \[\leadsto \frac{{\left(\frac{\color{blue}{{\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333} \cdot {\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow1/3 16.9%

         \[\leadsto \frac{{\left(\frac{\color{blue}{\sqrt[3]{\ell \cdot \sqrt{\frac{2}{\tan k}}}} \cdot {\left(\ell \cdot \sqrt{\frac{2}{\tan k}}\right)}^{0.3333333333333333}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow1/3 38.0%

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

   10. Simplified 38.0%

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

   if 1.0499999999999999e-20 < k < 3.39999999999999974e154

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0 75.4%

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

   if 3.39999999999999974e154 < k

   1. Initial program 29.3%

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

   3. Simplified 34.9%

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

      1. unpow3 34.9%

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

      2. *-un-lft-identity 34.9%

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

      3. times-frac 39.6%

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

      4. pow2 39.6%

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

   6. Applied egg-rr 39.6%

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

      1. /-rgt-identity 39.6%

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

      2. add-cube-cbrt 39.5%

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

      3. times-frac 41.9%

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

      4. unpow2 41.9%

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

      5. frac-times 58.2%

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

      6. unpow2 58.2%

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

   8. Applied egg-rr 58.2%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-20}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\ell \cdot \sqrt{\frac{2}{\tan k}}} \cdot \sqrt[3]{\ell \cdot \sqrt{\frac{2}{\tan k}}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\ell}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\ell}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-9)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (if (<= k_m 1.6e+154)
     (/
      2.0
      (/
       (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))
       (* (pow l 2.0) (cos k_m))))
     (/
      2.0
      (*
       (* (pow (/ t (cbrt l)) 2.0) (/ (/ t l) (cbrt l)))
       (* (+ 2.0 (pow (/ k_m t) 2.0)) (* (tan k_m) (sin k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-9) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 1.6e+154) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))) / (pow(l, 2.0) * cos(k_m)));
	} else {
		tmp = 2.0 / ((pow((t / cbrt(l)), 2.0) * ((t / l) / cbrt(l))) * ((2.0 + pow((k_m / t), 2.0)) * (tan(k_m) * sin(k_m))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-9) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 1.6e+154) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	} else {
		tmp = 2.0 / ((Math.pow((t / Math.cbrt(l)), 2.0) * ((t / l) / Math.cbrt(l))) * ((2.0 + Math.pow((k_m / t), 2.0)) * (Math.tan(k_m) * Math.sin(k_m))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-9)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	elseif (k_m <= 1.6e+154)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(t / cbrt(l)) ^ 2.0) * Float64(Float64(t / l) / cbrt(l))) * Float64(Float64(2.0 + (Float64(k_m / t) ^ 2.0)) * Float64(tan(k_m) * sin(k_m)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-9], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.6e+154], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.15e-9

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.6%

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

      3. pow2 39.6%

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

      4. sqrt-div 27.8%

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

      5. sqrt-pow1 29.6%

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

      6. metadata-eval 29.6%

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

      7. pow1 29.6%

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

      8. sqrt-prod 29.6%

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

      9. sqrt-pow1 34.7%

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

      10. metadata-eval 34.7%

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

      11. pow1 34.7%

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

      12. sqrt-pow1 38.4%

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

      13. metadata-eval 38.4%

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

   5. Applied egg-rr 38.4%

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

      1. *-lft-identity 38.4%

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

      2. associate-/r* 38.3%

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

   7. Simplified 38.3%

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

      1. div-inv 38.3%

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

      2. sqr-pow 38.3%

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

      3. times-frac 41.3%

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

      4. metadata-eval 41.3%

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

      5. metadata-eval 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. div-inv 41.3%

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

       3. pow-flip 41.3%

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

       4. metadata-eval 41.3%

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

   11. Applied egg-rr 41.3%

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

       1. *-lft-identity 41.3%

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

       2. associate-*l/ 41.3%

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

       3. *-lft-identity 41.3%

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

   13. Simplified 41.3%

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

   if 1.15e-9 < k < 1.6e154

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0 76.7%

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

   if 1.6e154 < k

   1. Initial program 29.3%

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

   3. Simplified 34.9%

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

      1. unpow3 34.9%

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

      2. *-un-lft-identity 34.9%

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

      3. times-frac 39.6%

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

      4. pow2 39.6%

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

   6. Applied egg-rr 39.6%

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

      1. /-rgt-identity 39.6%

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

      2. add-cube-cbrt 39.5%

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

      3. times-frac 41.9%

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

      4. unpow2 41.9%

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

      5. frac-times 58.2%

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

      6. unpow2 58.2%

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

   8. Applied egg-rr 58.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\ell}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 7.1 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.6e-9)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (if (<= k_m 7.1e+155)
     (/
      2.0
      (/
       (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))
       (* (pow l 2.0) (cos k_m))))
     (/
      2.0
      (*
       (* (+ 2.0 (pow (/ k_m t) 2.0)) (* (tan k_m) (sin k_m)))
       (pow (/ (/ t (cbrt l)) (cbrt l)) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.6e-9) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 7.1e+155) {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))) / (pow(l, 2.0) * cos(k_m)));
	} else {
		tmp = 2.0 / (((2.0 + pow((k_m / t), 2.0)) * (tan(k_m) * sin(k_m))) * pow(((t / cbrt(l)) / cbrt(l)), 3.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.6e-9) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 7.1e+155) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	} else {
		tmp = 2.0 / (((2.0 + Math.pow((k_m / t), 2.0)) * (Math.tan(k_m) * Math.sin(k_m))) * Math.pow(((t / Math.cbrt(l)) / Math.cbrt(l)), 3.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.6e-9)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	elseif (k_m <= 7.1e+155)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 + (Float64(k_m / t) ^ 2.0)) * Float64(tan(k_m) * sin(k_m))) * (Float64(Float64(t / cbrt(l)) / cbrt(l)) ^ 3.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.6e-9], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 7.1e+155], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.60000000000000006e-9

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.6%

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

      3. pow2 39.6%

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

      4. sqrt-div 27.8%

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

      5. sqrt-pow1 29.6%

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

      6. metadata-eval 29.6%

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

      7. pow1 29.6%

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

      8. sqrt-prod 29.6%

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

      9. sqrt-pow1 34.7%

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

      10. metadata-eval 34.7%

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

      11. pow1 34.7%

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

      12. sqrt-pow1 38.4%

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

      13. metadata-eval 38.4%

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

   5. Applied egg-rr 38.4%

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

      1. *-lft-identity 38.4%

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

      2. associate-/r* 38.3%

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

   7. Simplified 38.3%

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

      1. div-inv 38.3%

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

      2. sqr-pow 38.3%

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

      3. times-frac 41.3%

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

      4. metadata-eval 41.3%

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

      5. metadata-eval 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. div-inv 41.3%

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

       3. pow-flip 41.3%

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

       4. metadata-eval 41.3%

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

   11. Applied egg-rr 41.3%

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

       1. *-lft-identity 41.3%

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

       2. associate-*l/ 41.3%

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

       3. *-lft-identity 41.3%

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

   13. Simplified 41.3%

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

   if 1.60000000000000006e-9 < k < 7.09999999999999992e155

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0 76.7%

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

   if 7.09999999999999992e155 < k

   1. Initial program 29.3%

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

   3. Simplified 34.9%

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

      1. unpow3 34.9%

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

      2. *-un-lft-identity 34.9%

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

      3. times-frac 39.6%

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

      4. pow2 39.6%

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

   6. Applied egg-rr 39.6%

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

      1. add-cube-cbrt 39.4%

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

      2. pow3 39.4%

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

      3. cbrt-div 39.5%

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

      4. frac-times 34.9%

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

      5. unpow2 34.9%

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

      6. *-un-lft-identity 34.9%

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

      7. cbrt-div 34.9%

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

      8. add-cbrt-cube 58.0%

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

   8. Applied egg-rr 58.0%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k}\right)}^{2}\\ \mathbf{elif}\;k \leq 7.1 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.4e-9)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-9) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-9) then
        tmp = ((l / (t ** 0.75d0)) * ((t ** (-0.75d0)) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * (t * (sin(k_m) ** 2.0d0))) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-9) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.4e-9:
		tmp = math.pow(((l / math.pow(t, 0.75)) * (math.pow(t, -0.75) / k_m)), 2.0)
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * (t * math.pow(math.sin(k_m), 2.0))) / (math.pow(l, 2.0) * math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-9)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-9)
		tmp = ((l / (t ^ 0.75)) * ((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = 2.0 / (((k_m ^ 2.0) * (t * (sin(k_m) ^ 2.0))) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.4e-9], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e-9

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.6%

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

      3. pow2 39.6%

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

      4. sqrt-div 27.8%

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

      5. sqrt-pow1 29.6%

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

      6. metadata-eval 29.6%

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

      7. pow1 29.6%

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

      8. sqrt-prod 29.6%

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

      9. sqrt-pow1 34.7%

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

      10. metadata-eval 34.7%

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

      11. pow1 34.7%

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

      12. sqrt-pow1 38.4%

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

      13. metadata-eval 38.4%

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

   5. Applied egg-rr 38.4%

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

      1. *-lft-identity 38.4%

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

      2. associate-/r* 38.3%

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

   7. Simplified 38.3%

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

      1. div-inv 38.3%

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

      2. sqr-pow 38.3%

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

      3. times-frac 41.3%

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

      4. metadata-eval 41.3%

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

      5. metadata-eval 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. div-inv 41.3%

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

       3. pow-flip 41.3%

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

       4. metadata-eval 41.3%

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

   11. Applied egg-rr 41.3%

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

       1. *-lft-identity 41.3%

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

       2. associate-*l/ 41.3%

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

       3. *-lft-identity 41.3%

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

   13. Simplified 41.3%

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

   if 2.4e-9 < k

   1. Initial program 38.1%

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

   3. Taylor expanded in t around 0 60.4%

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

Alternative 5: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.8e-11)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (/
    2.0
    (*
     (/ (pow k_m 2.0) (pow l 2.0))
     (/ (* t (pow (sin k_m) 2.0)) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.8e-11) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) / pow(l, 2.0)) * ((t * pow(sin(k_m), 2.0)) / cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.8d-11) then
        tmp = ((l / (t ** 0.75d0)) * ((t ** (-0.75d0)) / k_m)) ** 2.0d0
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) / (l ** 2.0d0)) * ((t * (sin(k_m) ** 2.0d0)) / cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.8e-11) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) / Math.pow(l, 2.0)) * ((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.8e-11:
		tmp = math.pow(((l / math.pow(t, 0.75)) * (math.pow(t, -0.75) / k_m)), 2.0)
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) / math.pow(l, 2.0)) * ((t * math.pow(math.sin(k_m), 2.0)) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.8e-11)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.8e-11)
		tmp = ((l / (t ^ 0.75)) * ((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = 2.0 / (((k_m ^ 2.0) / (l ^ 2.0)) * ((t * (sin(k_m) ^ 2.0)) / cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.8e-11], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.8000000000000006e-11

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.6%

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

      3. pow2 39.6%

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

      4. sqrt-div 27.8%

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

      5. sqrt-pow1 29.6%

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

      6. metadata-eval 29.6%

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

      7. pow1 29.6%

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

      8. sqrt-prod 29.6%

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

      9. sqrt-pow1 34.7%

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

      10. metadata-eval 34.7%

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

      11. pow1 34.7%

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

      12. sqrt-pow1 38.4%

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

      13. metadata-eval 38.4%

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

   5. Applied egg-rr 38.4%

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

      1. *-lft-identity 38.4%

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

      2. associate-/r* 38.3%

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

   7. Simplified 38.3%

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

      1. div-inv 38.3%

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

      2. sqr-pow 38.3%

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

      3. times-frac 41.3%

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

      4. metadata-eval 41.3%

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

      5. metadata-eval 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. div-inv 41.3%

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

       3. pow-flip 41.3%

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

       4. metadata-eval 41.3%

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

   11. Applied egg-rr 41.3%

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

       1. *-lft-identity 41.3%

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

       2. associate-*l/ 41.3%

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

       3. *-lft-identity 41.3%

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

   13. Simplified 41.3%

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

   if 8.8000000000000006e-11 < k

   1. Initial program 38.1%

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

   3. Taylor expanded in k around inf 27.4%

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

      1. unpow2 27.4%

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

      2. unpow2 27.4%

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

      3. times-frac 35.5%

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

      4. unpow2 35.5%

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

   5. Simplified 35.5%

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

   6. Taylor expanded in t around 0 60.4%

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

      1. times-frac 55.3%

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

   8. Simplified 55.3%

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

Alternative 6: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \frac{\frac{{\ell}^{2} \cdot \cos k\_m}{{\sin k\_m}^{2}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-9)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (*
    (/ 2.0 (pow k_m 2.0))
    (/ (/ (* (pow l 2.0) (cos k_m)) (pow (sin k_m) 2.0)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-9) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = (2.0 / pow(k_m, 2.0)) * (((pow(l, 2.0) * cos(k_m)) / pow(sin(k_m), 2.0)) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.15d-9) then
        tmp = ((l / (t ** 0.75d0)) * ((t ** (-0.75d0)) / k_m)) ** 2.0d0
    else
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((((l ** 2.0d0) * cos(k_m)) / (sin(k_m) ** 2.0d0)) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-9) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (((Math.pow(l, 2.0) * Math.cos(k_m)) / Math.pow(Math.sin(k_m), 2.0)) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.15e-9:
		tmp = math.pow(((l / math.pow(t, 0.75)) * (math.pow(t, -0.75) / k_m)), 2.0)
	else:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (((math.pow(l, 2.0) * math.cos(k_m)) / math.pow(math.sin(k_m), 2.0)) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-9)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64(Float64(Float64((l ^ 2.0) * cos(k_m)) / (sin(k_m) ^ 2.0)) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.15e-9)
		tmp = ((l / (t ^ 0.75)) * ((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = (2.0 / (k_m ^ 2.0)) * ((((l ^ 2.0) * cos(k_m)) / (sin(k_m) ^ 2.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-9], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e-9

   1. Initial program 62.5%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.6%

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

      3. pow2 39.6%

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

      4. sqrt-div 27.8%

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

      5. sqrt-pow1 29.6%

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

      6. metadata-eval 29.6%

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

      7. pow1 29.6%

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

      8. sqrt-prod 29.6%

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

      9. sqrt-pow1 34.7%

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

      10. metadata-eval 34.7%

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

      11. pow1 34.7%

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

      12. sqrt-pow1 38.4%

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

      13. metadata-eval 38.4%

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

   5. Applied egg-rr 38.4%

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

      1. *-lft-identity 38.4%

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

      2. associate-/r* 38.3%

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

   7. Simplified 38.3%

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

      1. div-inv 38.3%

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

      2. sqr-pow 38.3%

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

      3. times-frac 41.3%

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

      4. metadata-eval 41.3%

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

      5. metadata-eval 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. *-un-lft-identity 41.3%

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

       2. div-inv 41.3%

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

       3. pow-flip 41.3%

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

       4. metadata-eval 41.3%

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

   11. Applied egg-rr 41.3%

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

       1. *-lft-identity 41.3%

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

       2. associate-*l/ 41.3%

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

       3. *-lft-identity 41.3%

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

   13. Simplified 41.3%

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

   if 1.15e-9 < k

   1. Initial program 38.1%

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

   3. Taylor expanded in t around 0 60.4%

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

      1. associate-*r/ 60.4%

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

      2. times-frac 55.2%

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

      3. times-frac 55.2%

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

      4. associate-*l/ 55.2%

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

      5. associate-/l* 55.2%

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

   5. Simplified 55.2%

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

Alternative 7: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00039:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 2.7 \cdot 10^{+96}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k\_m \cdot {t}^{3}}}{\tan k\_m}\right) \cdot \frac{\ell}{2 + {\left(\frac{k\_m}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00039)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (if (<= k_m 2.7e+96)
     (*
      (* l (/ (/ 2.0 (* (sin k_m) (pow t 3.0))) (tan k_m)))
      (/ l (+ 2.0 (pow (/ k_m t) 2.0))))
     0.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00039) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 2.7e+96) {
		tmp = (l * ((2.0 / (sin(k_m) * pow(t, 3.0))) / tan(k_m))) * (l / (2.0 + pow((k_m / t), 2.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.00039d0) then
        tmp = ((l / (t ** 0.75d0)) * ((t ** (-0.75d0)) / k_m)) ** 2.0d0
    else if (k_m <= 2.7d+96) then
        tmp = (l * ((2.0d0 / (sin(k_m) * (t ** 3.0d0))) / tan(k_m))) * (l / (2.0d0 + ((k_m / t) ** 2.0d0)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00039) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 2.7e+96) {
		tmp = (l * ((2.0 / (Math.sin(k_m) * Math.pow(t, 3.0))) / Math.tan(k_m))) * (l / (2.0 + Math.pow((k_m / t), 2.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.00039:
		tmp = math.pow(((l / math.pow(t, 0.75)) * (math.pow(t, -0.75) / k_m)), 2.0)
	elif k_m <= 2.7e+96:
		tmp = (l * ((2.0 / (math.sin(k_m) * math.pow(t, 3.0))) / math.tan(k_m))) * (l / (2.0 + math.pow((k_m / t), 2.0)))
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00039)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	elseif (k_m <= 2.7e+96)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / Float64(sin(k_m) * (t ^ 3.0))) / tan(k_m))) * Float64(l / Float64(2.0 + (Float64(k_m / t) ^ 2.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.00039)
		tmp = ((l / (t ^ 0.75)) * ((t ^ -0.75) / k_m)) ^ 2.0;
	elseif (k_m <= 2.7e+96)
		tmp = (l * ((2.0 / (sin(k_m) * (t ^ 3.0))) / tan(k_m))) * (l / (2.0 + ((k_m / t) ^ 2.0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00039], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 2.7e+96], N[(N[(l * N[(N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if k < 3.89999999999999993e-4

   1. Initial program 62.7%

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

   3. Taylor expanded in k around 0 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. add-sqr-sqrt 40.0%

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

      3. pow2 40.0%

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

      4. sqrt-div 28.2%

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

      5. sqrt-pow1 30.0%

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

      6. metadata-eval 30.0%

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

      7. pow1 30.0%

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

      8. sqrt-prod 30.0%

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

      9. sqrt-pow1 35.1%

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

      10. metadata-eval 35.1%

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

      11. pow1 35.1%

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

      12. sqrt-pow1 38.7%

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

      13. metadata-eval 38.7%

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

   5. Applied egg-rr 38.7%

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

      1. *-lft-identity 38.7%

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

      2. associate-/r* 38.6%

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

   7. Simplified 38.6%

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

      1. div-inv 38.6%

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

      2. sqr-pow 38.6%

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

      3. times-frac 41.6%

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

      4. metadata-eval 41.6%

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

      5. metadata-eval 41.6%

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

   9. Applied egg-rr 41.6%

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

       1. *-un-lft-identity 41.6%

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

       2. div-inv 41.6%

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

       3. pow-flip 41.6%

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

       4. metadata-eval 41.6%

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

   11. Applied egg-rr 41.6%

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

       1. *-lft-identity 41.6%

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

       2. associate-*l/ 41.6%

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

       3. *-lft-identity 41.6%

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

   13. Simplified 41.6%

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

   if 3.89999999999999993e-4 < k < 2.70000000000000022e96

   1. Initial program 62.4%

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

   3. Simplified 62.3%

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

      1. associate-*r* 62.5%

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

      2. *-un-lft-identity 62.5%

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

      3. times-frac 71.4%

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

      4. div-inv 71.4%

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

      5. frac-times 71.4%

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

      6. metadata-eval 71.4%

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

   6. Applied egg-rr 71.4%

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

      1. /-rgt-identity 71.4%

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

      2. *-commutative 71.4%

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

      3. *-rgt-identity 71.4%

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

      4. associate-/r* 71.4%

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

      5. associate-*l/ 71.4%

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

      6. associate-*r/ 71.4%

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

      7. associate-*l/ 71.5%

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

      8. associate-*r/ 71.5%

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

      9. metadata-eval 71.5%

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

      10. *-commutative 71.5%

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

   8. Simplified 71.5%

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

   if 2.70000000000000022e96 < k

   1. Initial program 27.1%

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

   3. Taylor expanded in k around 0 28.9%

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

      1. add-log-exp 29.0%

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

      2. add-sqr-sqrt 27.1%

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

      3. pow2 27.1%

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

      4. sqrt-div 11.7%

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

      5. sqrt-pow1 12.4%

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

      6. metadata-eval 12.4%

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

      7. pow1 12.4%

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

      8. sqrt-prod 12.4%

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

      9. sqrt-pow1 13.0%

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

      10. metadata-eval 13.0%

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

      11. pow1 13.0%

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

      12. sqrt-pow1 22.1%

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

      13. metadata-eval 22.1%

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

   5. Applied egg-rr 22.1%

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

   6. Taylor expanded in l around 0 52.1%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 52.1%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 52.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 12800:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 3.1 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 12800.0)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   (if (<= k_m 3.1e+96)
     (/ 2.0 (* 2.0 (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))))
     0.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 12800.0) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 3.1e+96) {
		tmp = 2.0 / (2.0 * (tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 12800.0d0) then
        tmp = ((l / (t ** 0.75d0)) * ((t ** (-0.75d0)) / k_m)) ** 2.0d0
    else if (k_m <= 3.1d+96) then
        tmp = 2.0d0 / (2.0d0 * (tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 12800.0) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else if (k_m <= 3.1e+96) {
		tmp = 2.0 / (2.0 * (Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 12800.0:
		tmp = math.pow(((l / math.pow(t, 0.75)) * (math.pow(t, -0.75) / k_m)), 2.0)
	elif k_m <= 3.1e+96:
		tmp = 2.0 / (2.0 * (math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))))
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 12800.0)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	elseif (k_m <= 3.1e+96)
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 12800.0)
		tmp = ((l / (t ^ 0.75)) * ((t ^ -0.75) / k_m)) ^ 2.0;
	elseif (k_m <= 3.1e+96)
		tmp = 2.0 / (2.0 * (tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 12800.0], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 3.1e+96], N[(2.0 / N[(2.0 * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if k < 12800

   1. Initial program 62.6%

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

   3. Taylor expanded in k around 0 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. add-sqr-sqrt 40.1%

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

      3. pow2 40.1%

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

      4. sqrt-div 28.5%

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

      5. sqrt-pow1 30.2%

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

      6. metadata-eval 30.2%

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

      7. pow1 30.2%

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

      8. sqrt-prod 30.2%

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

      9. sqrt-pow1 35.2%

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

      10. metadata-eval 35.2%

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

      11. pow1 35.2%

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

      12. sqrt-pow1 38.8%

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

      13. metadata-eval 38.8%

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

   5. Applied egg-rr 38.8%

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

      1. *-lft-identity 38.8%

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

      2. associate-/r* 38.7%

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

   7. Simplified 38.7%

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

      1. div-inv 38.7%

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

      2. sqr-pow 38.7%

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

      3. times-frac 41.7%

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

      4. metadata-eval 41.7%

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

      5. metadata-eval 41.7%

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

   9. Applied egg-rr 41.7%

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

       1. *-un-lft-identity 41.7%

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

       2. div-inv 41.7%

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

       3. pow-flip 41.7%

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

       4. metadata-eval 41.7%

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

   11. Applied egg-rr 41.7%

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

       1. *-lft-identity 41.7%

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

       2. associate-*l/ 41.7%

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

       3. *-lft-identity 41.7%

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

   13. Simplified 41.7%

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

   if 12800 < k < 3.0999999999999998e96

   1. Initial program 63.7%

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

   3. Taylor expanded in k around 0 59.5%

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

   if 3.0999999999999998e96 < k

   1. Initial program 27.1%

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

   3. Taylor expanded in k around 0 28.9%

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

      1. add-log-exp 29.0%

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

      2. add-sqr-sqrt 27.1%

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

      3. pow2 27.1%

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

      4. sqrt-div 11.7%

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

      5. sqrt-pow1 12.4%

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

      6. metadata-eval 12.4%

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

      7. pow1 12.4%

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

      8. sqrt-prod 12.4%

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

      9. sqrt-pow1 13.0%

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

      10. metadata-eval 13.0%

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

      11. pow1 13.0%

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

      12. sqrt-pow1 22.1%

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

      13. metadata-eval 22.1%

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

   5. Applied egg-rr 22.1%

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

   6. Taylor expanded in l around 0 52.1%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 52.1%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 52.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 12800:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k}\right)}^{2}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9 \cdot 10^{+93}:\\ \;\;\;\;{\left(\frac{\ell}{{t}^{0.75}} \cdot \frac{{t}^{-0.75}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9e+93)
   (pow (* (/ l (pow t 0.75)) (/ (pow t -0.75) k_m)) 2.0)
   0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e+93) {
		tmp = pow(((l / pow(t, 0.75)) * (pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9d+93) then
        tmp = ((l / (t ** 0.75d0)) * ((t ** (-0.75d0)) / k_m)) ** 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9e+93) {
		tmp = Math.pow(((l / Math.pow(t, 0.75)) * (Math.pow(t, -0.75) / k_m)), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9e+93:
		tmp = math.pow(((l / math.pow(t, 0.75)) * (math.pow(t, -0.75) / k_m)), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9e+93)
		tmp = Float64(Float64(l / (t ^ 0.75)) * Float64((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9e+93)
		tmp = ((l / (t ^ 0.75)) * ((t ^ -0.75) / k_m)) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9e+93], N[Power[N[(N[(l / N[Power[t, 0.75], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, -0.75], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 8.99999999999999981e93

   1. Initial program 62.9%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.5%

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

      3. pow2 39.5%

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

      4. sqrt-div 27.5%

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

      5. sqrt-pow1 28.6%

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

      6. metadata-eval 28.6%

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

      7. pow1 28.6%

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

      8. sqrt-prod 28.6%

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

      9. sqrt-pow1 33.2%

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

      10. metadata-eval 33.2%

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

      11. pow1 33.2%

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

      12. sqrt-pow1 36.4%

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

      13. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

      1. *-lft-identity 36.4%

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

      2. associate-/r* 36.4%

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

   7. Simplified 36.4%

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

      1. div-inv 36.4%

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

      2. sqr-pow 36.4%

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

      3. times-frac 39.1%

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

      4. metadata-eval 39.1%

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

      5. metadata-eval 39.1%

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

   9. Applied egg-rr 39.1%

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

       1. *-un-lft-identity 39.1%

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

       2. div-inv 39.1%

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

       3. pow-flip 39.1%

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

       4. metadata-eval 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. *-lft-identity 39.1%

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

       2. associate-*l/ 39.1%

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

       3. *-lft-identity 39.1%

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

   13. Simplified 39.1%

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

   if 8.99999999999999981e93 < k

   1. Initial program 26.7%

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

   3. Taylor expanded in k around 0 28.4%

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

      1. add-log-exp 28.5%

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

      2. add-sqr-sqrt 26.6%

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

      3. pow2 26.6%

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

      4. sqrt-div 11.5%

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

      5. sqrt-pow1 12.2%

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

      6. metadata-eval 12.2%

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

      7. pow1 12.2%

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

      8. sqrt-prod 12.2%

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

      9. sqrt-pow1 12.8%

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

      10. metadata-eval 12.8%

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

      11. pow1 12.8%

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

      12. sqrt-pow1 21.7%

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

      13. metadata-eval 21.7%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in l around 0 51.2%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 51.2%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{+94}:\\ \;\;\;\;{\left(\frac{1}{k\_m} \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e+94) (pow (* (/ 1.0 k_m) (/ l (pow t 1.5))) 2.0) 0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+94) {
		tmp = pow(((1.0 / k_m) * (l / pow(t, 1.5))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d+94) then
        tmp = ((1.0d0 / k_m) * (l / (t ** 1.5d0))) ** 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+94) {
		tmp = Math.pow(((1.0 / k_m) * (l / Math.pow(t, 1.5))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25e+94:
		tmp = math.pow(((1.0 / k_m) * (l / math.pow(t, 1.5))), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e+94)
		tmp = Float64(Float64(1.0 / k_m) * Float64(l / (t ^ 1.5))) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25e+94)
		tmp = ((1.0 / k_m) * (l / (t ^ 1.5))) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e+94], N[Power[N[(N[(1.0 / k$95$m), $MachinePrecision] * N[(l / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000003e94

   1. Initial program 62.9%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.5%

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

      3. pow2 39.5%

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

      4. sqrt-div 27.5%

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

      5. sqrt-pow1 28.6%

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

      6. metadata-eval 28.6%

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

      7. pow1 28.6%

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

      8. sqrt-prod 28.6%

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

      9. sqrt-pow1 33.2%

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

      10. metadata-eval 33.2%

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

      11. pow1 33.2%

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

      12. sqrt-pow1 36.4%

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

      13. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

      1. *-lft-identity 36.4%

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

      2. associate-/r* 36.4%

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

   7. Simplified 36.4%

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

      1. associate-/r* 36.4%

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

      2. *-un-lft-identity 36.4%

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

      3. times-frac 36.5%

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

   9. Applied egg-rr 36.5%

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

   if 1.25000000000000003e94 < k

   1. Initial program 26.7%

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

   3. Taylor expanded in k around 0 28.4%

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

      1. add-log-exp 28.5%

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

      2. add-sqr-sqrt 26.6%

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

      3. pow2 26.6%

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

      4. sqrt-div 11.5%

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

      5. sqrt-pow1 12.2%

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

      6. metadata-eval 12.2%

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

      7. pow1 12.2%

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

      8. sqrt-prod 12.2%

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

      9. sqrt-pow1 12.8%

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

      10. metadata-eval 12.8%

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

      11. pow1 12.8%

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

      12. sqrt-pow1 21.7%

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

      13. metadata-eval 21.7%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in l around 0 51.2%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 51.2%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 45.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{+94}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m \cdot {t}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e+94) (pow (/ l (* k_m (pow t 1.5))) 2.0) 0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e+94) {
		tmp = pow((l / (k_m * pow(t, 1.5))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.3d+94) then
        tmp = (l / (k_m * (t ** 1.5d0))) ** 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e+94) {
		tmp = Math.pow((l / (k_m * Math.pow(t, 1.5))), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.3e+94:
		tmp = math.pow((l / (k_m * math.pow(t, 1.5))), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e+94)
		tmp = Float64(l / Float64(k_m * (t ^ 1.5))) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.3e+94)
		tmp = (l / (k_m * (t ^ 1.5))) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e+94], N[Power[N[(l / N[(k$95$m * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e94

   1. Initial program 62.9%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. sqrt-div 27.5%

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

      4. sqrt-pow1 28.6%

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

      5. metadata-eval 28.6%

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

      6. pow1 28.6%

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

      7. sqrt-prod 28.6%

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

      8. sqrt-pow1 33.2%

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

      9. metadata-eval 33.2%

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

      10. pow1 33.2%

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

      11. sqrt-pow1 36.4%

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

      12. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

   if 1.3e94 < k

   1. Initial program 26.7%

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

   3. Taylor expanded in k around 0 28.4%

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

      1. add-log-exp 28.5%

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

      2. add-sqr-sqrt 26.6%

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

      3. pow2 26.6%

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

      4. sqrt-div 11.5%

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

      5. sqrt-pow1 12.2%

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

      6. metadata-eval 12.2%

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

      7. pow1 12.2%

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

      8. sqrt-prod 12.2%

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

      9. sqrt-pow1 12.8%

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

      10. metadata-eval 12.8%

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

      11. pow1 12.8%

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

      12. sqrt-pow1 21.7%

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

      13. metadata-eval 21.7%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in l around 0 51.2%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 51.2%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{+94}:\\ \;\;\;\;{\left({t}^{-1.5} \cdot \frac{\ell}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e+94) (pow (* (pow t -1.5) (/ l k_m)) 2.0) 0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e+94) {
		tmp = pow((pow(t, -1.5) * (l / k_m)), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.3d+94) then
        tmp = ((t ** (-1.5d0)) * (l / k_m)) ** 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e+94) {
		tmp = Math.pow((Math.pow(t, -1.5) * (l / k_m)), 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.3e+94:
		tmp = math.pow((math.pow(t, -1.5) * (l / k_m)), 2.0)
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e+94)
		tmp = Float64((t ^ -1.5) * Float64(l / k_m)) ^ 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.3e+94)
		tmp = ((t ^ -1.5) * (l / k_m)) ^ 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e+94], N[Power[N[(N[Power[t, -1.5], $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e94

   1. Initial program 62.9%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.5%

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

      3. pow2 39.5%

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

      4. sqrt-div 27.5%

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

      5. sqrt-pow1 28.6%

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

      6. metadata-eval 28.6%

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

      7. pow1 28.6%

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

      8. sqrt-prod 28.6%

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

      9. sqrt-pow1 33.2%

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

      10. metadata-eval 33.2%

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

      11. pow1 33.2%

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

      12. sqrt-pow1 36.4%

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

      13. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

      1. *-lft-identity 36.4%

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

      2. associate-/r* 36.4%

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

   7. Simplified 36.4%

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

      1. associate-/l/ 36.4%

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

      2. *-un-lft-identity 36.4%

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

      3. times-frac 36.4%

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

      4. pow-flip 36.4%

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

      5. metadata-eval 36.4%

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

   9. Applied egg-rr 36.4%

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

   if 1.3e94 < k

   1. Initial program 26.7%

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

   3. Taylor expanded in k around 0 28.4%

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

      1. add-log-exp 28.5%

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

      2. add-sqr-sqrt 26.6%

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

      3. pow2 26.6%

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

      4. sqrt-div 11.5%

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

      5. sqrt-pow1 12.2%

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

      6. metadata-eval 12.2%

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

      7. pow1 12.2%

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

      8. sqrt-prod 12.2%

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

      9. sqrt-pow1 12.8%

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

      10. metadata-eval 12.8%

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

      11. pow1 12.8%

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

      12. sqrt-pow1 21.7%

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

      13. metadata-eval 21.7%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in l around 0 51.2%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 51.2%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 66.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m}}{\frac{{\left(-t\right)}^{3}}{\frac{-\ell}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e+94) (/ (/ l k_m) (/ (pow (- t) 3.0) (/ (- l) k_m))) 0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+94) {
		tmp = (l / k_m) / (pow(-t, 3.0) / (-l / k_m));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d+94) then
        tmp = (l / k_m) / ((-t ** 3.0d0) / (-l / k_m))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+94) {
		tmp = (l / k_m) / (Math.pow(-t, 3.0) / (-l / k_m));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25e+94:
		tmp = (l / k_m) / (math.pow(-t, 3.0) / (-l / k_m))
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e+94)
		tmp = Float64(Float64(l / k_m) / Float64((Float64(-t) ^ 3.0) / Float64(Float64(-l) / k_m)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25e+94)
		tmp = (l / k_m) / ((-t ^ 3.0) / (-l / k_m));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e+94], N[(N[(l / k$95$m), $MachinePrecision] / N[(N[Power[(-t), 3.0], $MachinePrecision] / N[((-l) / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000003e94

   1. Initial program 62.9%

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

   3. Taylor expanded in k around 0 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. add-sqr-sqrt 39.5%

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

      3. pow2 39.5%

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

      4. sqrt-div 27.5%

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

      5. sqrt-pow1 28.6%

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

      6. metadata-eval 28.6%

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

      7. pow1 28.6%

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

      8. sqrt-prod 28.6%

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

      9. sqrt-pow1 33.2%

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

      10. metadata-eval 33.2%

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

      11. pow1 33.2%

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

      12. sqrt-pow1 36.4%

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

      13. metadata-eval 36.4%

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

   5. Applied egg-rr 36.4%

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

      1. *-lft-identity 36.4%

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

      2. associate-/r* 36.4%

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

   7. Simplified 36.4%

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

      1. unpow2 36.4%

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

      2. associate-/r* 36.3%

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

      3. clear-num 36.3%

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

      4. frac-2neg 36.3%

         \[\leadsto \frac{1}{\frac{k \cdot {t}^{1.5}}{\ell}} \cdot \color{blue}{\frac{-\frac{\ell}{k}}{-{t}^{1.5}}} \]

      5. frac-times 36.4%

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

   9. Applied egg-rr 36.4%

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

       1. *-lft-identity 36.4%

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

       2. *-commutative 36.4%

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

       3. associate-/r* 36.4%

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

       4. *-commutative 36.4%

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

       5. associate-*l/ 36.4%

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

       6. associate-/r/ 36.4%

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

       7. associate-/r* 36.4%

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

       8. distribute-neg-frac2 36.4%

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

       9. associate-*r/ 33.0%

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

       10. distribute-lft-neg-out 33.0%

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

       11. pow-sqr 66.6%

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

       12. metadata-eval 66.6%

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

       13. cube-mult 66.6%

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

       14. unpow2 66.6%

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

       15. distribute-lft-neg-out 66.6%

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

       16. unpow2 66.6%

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

       17. sqr-neg 66.6%

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

       18. cube-unmult 66.6%

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

   11. Simplified 66.6%

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

   if 1.25000000000000003e94 < k

   1. Initial program 26.7%

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

   3. Taylor expanded in k around 0 28.4%

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

      1. add-log-exp 28.5%

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

      2. add-sqr-sqrt 26.6%

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

      3. pow2 26.6%

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

      4. sqrt-div 11.5%

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

      5. sqrt-pow1 12.2%

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

      6. metadata-eval 12.2%

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

      7. pow1 12.2%

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

      8. sqrt-prod 12.2%

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

      9. sqrt-pow1 12.8%

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

      10. metadata-eval 12.8%

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

      11. pow1 12.8%

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

      12. sqrt-pow1 21.7%

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

      13. metadata-eval 21.7%

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

   5. Applied egg-rr 21.7%

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

   6. Taylor expanded in l around 0 51.2%

      \[\leadsto \log \color{blue}{1} \]
   7. Step-by-step derivation

      1. metadata-eval 51.2%

         \[\leadsto \color{blue}{0} \]

   8. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{\left(-t\right)}^{3}}{\frac{-\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.2% accurate, 421.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 0 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 0.0)

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 0.0;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 0.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 0.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 0.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return 0.0
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 0.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := 0.0

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in k around 0 48.3%

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

   1. add-log-exp 46.9%

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

   2. add-sqr-sqrt 36.1%

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

   3. pow2 36.1%

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

   4. sqrt-div 23.4%

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

   5. sqrt-pow1 24.9%

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

   6. metadata-eval 24.9%

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

   7. pow1 24.9%

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

   8. sqrt-prod 24.8%

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

   9. sqrt-pow1 28.2%

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

   10. metadata-eval 28.2%

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

   11. pow1 28.2%

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

   12. sqrt-pow1 31.5%

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

   13. metadata-eval 31.5%

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

5. Applied egg-rr 31.5%

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

6. Taylor expanded in l around 0 39.5%

   \[\leadsto \log \color{blue}{1} \]
7. Step-by-step derivation

   1. metadata-eval 39.5%

      \[\leadsto \color{blue}{0} \]

8. Applied egg-rr 39.5%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024166 
(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))))