Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.0% → 93.8%

Time: 18.5s

Alternatives: 13

Speedup: 17.1×

• 38.8% 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: 35.0% 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: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.55 \cdot 10^{-21}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{\ell + \ell}{\sin k\_m \cdot \left(\sin k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.55e-21)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (*
    (/ (* l (cos k_m)) k_m)
    (/ (+ l l) (* (sin k_m) (* (sin k_m) (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.55e-21) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l * cos(k_m)) / k_m) * ((l + l) / (sin(k_m) * (sin(k_m) * (k_m * 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 <= 2.55d-21) then
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
    else
        tmp = ((l * cos(k_m)) / k_m) * ((l + l) / (sin(k_m) * (sin(k_m) * (k_m * 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 <= 2.55e-21) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l * Math.cos(k_m)) / k_m) * ((l + l) / (Math.sin(k_m) * (Math.sin(k_m) * (k_m * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.55e-21:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	else:
		tmp = ((l * math.cos(k_m)) / k_m) * ((l + l) / (math.sin(k_m) * (math.sin(k_m) * (k_m * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.55e-21)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(l + l) / Float64(sin(k_m) * Float64(sin(k_m) * Float64(k_m * t)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.55e-21)
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	else
		tmp = ((l * cos(k_m)) / k_m) * ((l + l) / (sin(k_m) * (sin(k_m) * (k_m * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.55e-21], N[(N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.55000000000000002e-21

   1. Initial program 41.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 90.0%

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

   if 2.55000000000000002e-21 < k

   1. Initial program 31.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 91.8%

      \[\leadsto \frac{\ell \cdot \cos k}{k} \cdot \color{blue}{\frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.8%

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

   11. Applied rewrites 94.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-21}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{\ell + \ell}{\sin k \cdot \left(\sin k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{\ell + \ell}{{\sin k\_m}^{2} \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e-10)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (* (/ (* l (cos k_m)) k_m) (/ (+ l l) (* (pow (sin k_m) 2.0) (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-10) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l * cos(k_m)) / k_m) * ((l + l) / (pow(sin(k_m), 2.0) * (k_m * 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 <= 7.5d-10) then
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
    else
        tmp = ((l * cos(k_m)) / k_m) * ((l + l) / ((sin(k_m) ** 2.0d0) * (k_m * 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 <= 7.5e-10) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l * Math.cos(k_m)) / k_m) * ((l + l) / (Math.pow(Math.sin(k_m), 2.0) * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.5e-10:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	else:
		tmp = ((l * math.cos(k_m)) / k_m) * ((l + l) / (math.pow(math.sin(k_m), 2.0) * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e-10)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(l + l) / Float64((sin(k_m) ^ 2.0) * Float64(k_m * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.5e-10)
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	else
		tmp = ((l * cos(k_m)) / k_m) * ((l + l) / ((sin(k_m) ^ 2.0) * (k_m * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.5e-10], N[(N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999995e-10

   1. Initial program 40.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 93.3%

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

   if 7.49999999999999995e-10 < k

   1. Initial program 29.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 93.1%

      \[\leadsto \frac{\ell \cdot \cos k}{k} \cdot \color{blue}{\frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{\ell + \ell}{{\sin k}^{2} \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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))))