Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 97.0%

Time: 16.2s

Alternatives: 16

Speedup: 25.6×

• 40.2% 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: 36.0% 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: 97.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-100}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m} \cdot \frac{\ell \cdot \frac{1}{k\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.05e-100)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (* (/ (+ l l) k_m) (/ (* l (/ 1.0 (* k_m (* (sin k_m) (tan k_m))))) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.05e-100) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l + l) / k_m) * ((l * (1.0 / (k_m * (sin(k_m) * tan(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 <= 1.05d-100) then
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
    else
        tmp = ((l + l) / k_m) * ((l * (1.0d0 / (k_m * (sin(k_m) * tan(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 <= 1.05e-100) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l + l) / k_m) * ((l * (1.0 / (k_m * (Math.sin(k_m) * Math.tan(k_m))))) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.05e-100:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	else:
		tmp = ((l + l) / k_m) * ((l * (1.0 / (k_m * (math.sin(k_m) * math.tan(k_m))))) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.05e-100)
		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 + l) / k_m) * Float64(Float64(l * Float64(1.0 / Float64(k_m * Float64(sin(k_m) * tan(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 <= 1.05e-100)
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	else
		tmp = ((l + l) / k_m) * ((l * (1.0 / (k_m * (sin(k_m) * tan(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, 1.05e-100], 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 + l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[(1.0 / N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05000000000000005e-100

   1. Initial program 43.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 64.8

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

   5. Simplified 64.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 81.6

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

   7. Applied egg-rr 81.6%

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

   if 1.05000000000000005e-100 < k

   1. Initial program 24.5%

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

   3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 76.6

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

   5. Simplified 76.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 79.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\frac{1}{\sin k \cdot \tan k}}{k}}}{t} \cdot \frac{\ell + \ell}{k} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sin k \cdot \tan k}}{k} \cdot \ell}}{t} \cdot \frac{\ell + \ell}{k} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sin k \cdot \tan k}}{k} \cdot \ell}}{t} \cdot \frac{\ell + \ell}{k} \]

       4. associate-/l/ N/A

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

       5. /-lowering-/.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. sin-lowering-sin.f64 N/A

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

       9. tan-lowering-tan.f64 98.5

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

   11. Applied egg-rr 98.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-100}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\ell \cdot \frac{1}{k \cdot \left(\sin k \cdot \tan k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + -1\right)} \leq -4 \cdot 10^{-181}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m} \cdot \frac{\frac{\frac{\mathsf{fma}\left(\ell, \left(k\_m \cdot k\_m\right) \cdot -0.16666666666666666, \ell\right)}{k\_m \cdot k\_m}}{k\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) -1.0)))
      -4e-181)
   (*
    (/ (+ l l) k_m)
    (/
     (/ (/ (fma l (* (* k_m k_m) -0.16666666666666666) l) (* k_m k_m)) k_m)
     t))
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * ((1.0 + pow((k_m / t), 2.0)) + -1.0))) <= -4e-181) {
		tmp = ((l + l) / k_m) * (((fma(l, ((k_m * k_m) * -0.16666666666666666), l) / (k_m * k_m)) / k_m) / t);
	} else {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + -1.0))) <= -4e-181)
		tmp = Float64(Float64(Float64(l + l) / k_m) * Float64(Float64(Float64(fma(l, Float64(Float64(k_m * k_m) * -0.16666666666666666), l) / Float64(k_m * k_m)) / k_m) / t));
	else
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(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] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-181], N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(N[(l * N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -4.00000000000000019e-181

   1. Initial program 90.4%

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

   3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 97.5

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

   5. Simplified 97.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 90.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]

   10. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{\ell + \frac{-1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{6} \cdot \left({k}^{2} \cdot \ell\right) + \ell}}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\frac{-1}{6} \cdot {k}^{2}\right) \cdot \ell} + \ell}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \left(\frac{-1}{6} \cdot {k}^{2}\right)} + \ell}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot {k}^{2}, \ell\right)}}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \color{blue}{\frac{-1}{6} \cdot {k}^{2}}, \ell\right)}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot \color{blue}{\left(k \cdot k\right)}, \ell\right)}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot \color{blue}{\left(k \cdot k\right)}, \ell\right)}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot \left(k \cdot k\right), \ell\right)}{\color{blue}{k \cdot k}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       10. *-lowering-*.f64 88.8

           \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, -0.16666666666666666 \cdot \left(k \cdot k\right), \ell\right)}{\color{blue}{k \cdot k}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

   12. Simplified 88.8%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\ell, -0.16666666666666666 \cdot \left(k \cdot k\right), \ell\right)}{k \cdot k}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

   if -4.00000000000000019e-181 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.8%

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

   3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 60.3

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

   5. Simplified 60.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 75.2

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

   7. Applied egg-rr 75.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq -4 \cdot 10^{-181}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\frac{\frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.16666666666666666, \ell\right)}{k \cdot k}}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) -1.0)))
      -2e-76)
   (/ -1.0 0.0)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * ((1.0 + pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * 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 ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + (-1.0d0)))) <= (-2d-76)) then
        tmp = (-1.0d0) / 0.0d0
    else
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * 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 ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * ((1.0 + Math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * ((1.0 + math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76:
		tmp = -1.0 / 0.0
	else:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = Float64(-1.0 / 0.0);
	else
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * ((1.0 + ((k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = -1.0 / 0.0;
	else
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(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] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-76], N[(-1.0 / 0.0), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -1.99999999999999985e-76

   1. Initial program 90.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 80.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]

   if -1.99999999999999985e-76 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 28.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 74.9

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

   7. Applied egg-rr 74.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m} \cdot \frac{\ell}{k\_m \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) -1.0)))
      -2e-76)
   (/ -1.0 0.0)
   (* (/ (+ l l) k_m) (/ l (* k_m (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * ((1.0 + pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = ((l + l) / k_m) * (l / (k_m * (t * (k_m * 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 ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + (-1.0d0)))) <= (-2d-76)) then
        tmp = (-1.0d0) / 0.0d0
    else
        tmp = ((l + l) / k_m) * (l / (k_m * (t * (k_m * 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 ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * ((1.0 + Math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = ((l + l) / k_m) * (l / (k_m * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * ((1.0 + math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76:
		tmp = -1.0 / 0.0
	else:
		tmp = ((l + l) / k_m) * (l / (k_m * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = Float64(-1.0 / 0.0);
	else
		tmp = Float64(Float64(Float64(l + l) / k_m) * Float64(l / Float64(k_m * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * ((1.0 + ((k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = -1.0 / 0.0;
	else
		tmp = ((l + l) / k_m) * (l / (k_m * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(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] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-76], N[(-1.0 / 0.0), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(k$95$m * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -1.99999999999999985e-76

   1. Initial program 90.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 80.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]

   if -1.99999999999999985e-76 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 28.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 73.6

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

   5. Simplified 73.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell + \ell}{k}} \]

   8. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 74.0

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

   10. Simplified 74.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell + \ell}{k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) -1.0)))
      -2e-76)
   (/ -1.0 0.0)
   (* l (/ (+ l l) (* k_m (* t (* k_m (* k_m k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * ((1.0 + pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = l * ((l + l) / (k_m * (t * (k_m * (k_m * 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 ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + (-1.0d0)))) <= (-2d-76)) then
        tmp = (-1.0d0) / 0.0d0
    else
        tmp = l * ((l + l) / (k_m * (t * (k_m * (k_m * 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 ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * ((1.0 + Math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = l * ((l + l) / (k_m * (t * (k_m * (k_m * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * ((1.0 + math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76:
		tmp = -1.0 / 0.0
	else:
		tmp = l * ((l + l) / (k_m * (t * (k_m * (k_m * k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = Float64(-1.0 / 0.0);
	else
		tmp = Float64(l * Float64(Float64(l + l) / Float64(k_m * Float64(t * Float64(k_m * Float64(k_m * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * ((1.0 + ((k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = -1.0 / 0.0;
	else
		tmp = l * ((l + l) / (k_m * (t * (k_m * (k_m * k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(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] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-76], N[(-1.0 / 0.0), $MachinePrecision], N[(l * N[(N[(l + l), $MachinePrecision] / N[(k$95$m * N[(t * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -1.99999999999999985e-76

   1. Initial program 90.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 80.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]

   if -1.99999999999999985e-76 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 28.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

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

      4. distribute-lft-out-- N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      6. +-inverses N/A

         \[\leadsto \frac{\frac{\ell \cdot 0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \frac{0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      8. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 70.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell + \ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) -1.0)))
      -2e-76)
   (/ -1.0 0.0)
   (/ (+ l l) (* t (* (* k_m k_m) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * ((1.0 + pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = (l + l) / (t * ((k_m * k_m) * (k_m * 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 ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + (-1.0d0)))) <= (-2d-76)) then
        tmp = (-1.0d0) / 0.0d0
    else
        tmp = (l + l) / (t * ((k_m * k_m) * (k_m * 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 ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * ((1.0 + Math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = (l + l) / (t * ((k_m * k_m) * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * ((1.0 + math.pow((k_m / t), 2.0)) + -1.0))) <= -2e-76:
		tmp = -1.0 / 0.0
	else:
		tmp = (l + l) / (t * ((k_m * k_m) * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = Float64(-1.0 / 0.0);
	else
		tmp = Float64(Float64(l + l) / Float64(t * Float64(Float64(k_m * k_m) * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * ((1.0 + ((k_m / t) ^ 2.0)) + -1.0))) <= -2e-76)
		tmp = -1.0 / 0.0;
	else
		tmp = (l + l) / (t * ((k_m * k_m) * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(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] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-76], N[(-1.0 / 0.0), $MachinePrecision], N[(N[(l + l), $MachinePrecision] / N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -1.99999999999999985e-76

   1. Initial program 90.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 80.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]

   if -1.99999999999999985e-76 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 28.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

      1. flip-+ N/A

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

      2. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      3. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. +-inverses N/A

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\ell - \ell\right)}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      7. +-inverses N/A

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

      8. flip-+ N/A

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

      9. +-lowering-+.f64 44.1

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

   7. Applied egg-rr 44.1%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + -1\right)} \leq -1.5:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t 3.0) (* l l))))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) -1.0)))
      -1.5)
   (/ -1.0 0.0)
   0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t, 3.0) / (l * l)))) * ((1.0 + pow((k_m / t), 2.0)) + -1.0))) <= -1.5) {
		tmp = -1.0 / 0.0;
	} 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 ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t ** 3.0d0) / (l * l)))) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + (-1.0d0)))) <= (-1.5d0)) then
        tmp = (-1.0d0) / 0.0d0
    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 ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l)))) * ((1.0 + Math.pow((k_m / t), 2.0)) + -1.0))) <= -1.5) {
		tmp = -1.0 / 0.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l)))) * ((1.0 + math.pow((k_m / t), 2.0)) + -1.0))) <= -1.5:
		tmp = -1.0 / 0.0
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + -1.0))) <= -1.5)
		tmp = Float64(-1.0 / 0.0);
	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 ((2.0 / ((tan(k_m) * (sin(k_m) * ((t ^ 3.0) / (l * l)))) * ((1.0 + ((k_m / t) ^ 2.0)) + -1.0))) <= -1.5)
		tmp = -1.0 / 0.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -1.5

   1. Initial program 89.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 89.7

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

   4. Applied egg-rr 89.7%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 84.5%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 84.5%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]

   if -1.5 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 28.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 42.6

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

   4. Applied egg-rr 42.6%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 29.3%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{0}{-1}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{0}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{0 - 0}} \]

       4. flip-- N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0} - 0 \cdot 0}{0 + 0}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{0}}{0 + 0}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0}}{0 + 0}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{0}{\color{blue}{0}}} \]

       9. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k - k}}{0}} \]

       10. +-inverses N/A

           \[\leadsto \frac{1}{\frac{k - k}{\color{blue}{k \cdot k - k \cdot k}}} \]

       11. clear-num N/A

           \[\leadsto \color{blue}{\frac{k \cdot k - k \cdot k}{k - k}} \]

       12. +-inverses N/A

           \[\leadsto \frac{\color{blue}{0}}{k - k} \]

       13. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{0 - 0}}{k - k} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{0 \cdot 0} - 0}{k - k} \]

       15. metadata-eval N/A

           \[\leadsto \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{k - k} \]

       16. +-inverses N/A

           \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

       18. flip-- N/A

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

       19. metadata-eval 30.6

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

   11. Applied egg-rr 30.6%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq -1.5:\\ \;\;\;\;\frac{-1}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\ell \cdot \frac{1}{\sin k\_m \cdot \tan k\_m}}{k\_m}}{t} \cdot \frac{\ell + \ell}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ (* l (/ 1.0 (* (sin k_m) (tan k_m)))) k_m) t) (/ (+ l l) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((l * (1.0 / (sin(k_m) * tan(k_m)))) / k_m) / t) * ((l + l) / k_m);
}

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 = (((l * (1.0d0 / (sin(k_m) * tan(k_m)))) / k_m) / t) * ((l + l) / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((l * (1.0 / (Math.sin(k_m) * Math.tan(k_m)))) / k_m) / t) * ((l + l) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((l * (1.0 / (math.sin(k_m) * math.tan(k_m)))) / k_m) / t) * ((l + l) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l * Float64(1.0 / Float64(sin(k_m) * tan(k_m)))) / k_m) / t) * Float64(Float64(l + l) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((l * (1.0 / (sin(k_m) * tan(k_m)))) / k_m) / t) * ((l + l) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l * N[(1.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 77.2

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

5. Simplified 77.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

7. Applied egg-rr 83.9%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

9. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]
10. Add Preprocessing

Alternative 9: 97.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell + \ell}{k\_m} \cdot \frac{\frac{\ell \cdot \frac{1}{k\_m}}{\sin k\_m \cdot \tan k\_m}}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (+ l l) k_m) (/ (/ (* l (/ 1.0 k_m)) (* (sin k_m) (tan k_m))) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l + l) / k_m) * (((l * (1.0 / k_m)) / (sin(k_m) * tan(k_m))) / t);
}

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 = ((l + l) / k_m) * (((l * (1.0d0 / k_m)) / (sin(k_m) * tan(k_m))) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l + l) / k_m) * (((l * (1.0 / k_m)) / (Math.sin(k_m) * Math.tan(k_m))) / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l + l) / k_m) * (((l * (1.0 / k_m)) / (math.sin(k_m) * math.tan(k_m))) / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l + l) / k_m) * Float64(Float64(Float64(l * Float64(1.0 / k_m)) / Float64(sin(k_m) * tan(k_m))) / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l + l) / k_m) * (((l * (1.0 / k_m)) / (sin(k_m) * tan(k_m))) / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l * N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 77.2

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

5. Simplified 77.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

7. Applied egg-rr 83.9%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

9. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]
10. Step-by-step derivation

    1. div-inv N/A

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

    2. un-div-inv N/A

       \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot \tan k}} \cdot \frac{1}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

    3. associate-*l/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \frac{1}{k}}{\sin k \cdot \tan k}}}{t} \cdot \frac{\ell + \ell}{k} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \frac{1}{k}}{\sin k \cdot \tan k}}}{t} \cdot \frac{\ell + \ell}{k} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \frac{1}{k}}}{\sin k \cdot \tan k}}{t} \cdot \frac{\ell + \ell}{k} \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\frac{1}{k}}}{\sin k \cdot \tan k}}{t} \cdot \frac{\ell + \ell}{k} \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \frac{1}{k}}{\color{blue}{\sin k \cdot \tan k}}}{t} \cdot \frac{\ell + \ell}{k} \]

    8. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{\frac{\ell \cdot \frac{1}{k}}{\color{blue}{\sin k} \cdot \tan k}}{t} \cdot \frac{\ell + \ell}{k} \]

    9. tan-lowering-tan.f64 96.5

       \[\leadsto \frac{\frac{\ell \cdot \frac{1}{k}}{\sin k \cdot \color{blue}{\tan k}}}{t} \cdot \frac{\ell + \ell}{k} \]

11. Applied egg-rr 96.5%

    \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \frac{1}{k}}{\sin k \cdot \tan k}}}{t} \cdot \frac{\ell + \ell}{k} \]

12. Final simplification 96.5%

    \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\frac{\ell \cdot \frac{1}{k}}{\sin k \cdot \tan k}}{t} \]
13. Add Preprocessing

Alternative 10: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k\_m} \cdot \frac{\frac{\ell}{k\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.6e-88)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (* (/ (+ l l) k_m) (/ (/ l (* k_m (* (sin k_m) (tan k_m)))) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.6e-88) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l + l) / k_m) * ((l / (k_m * (sin(k_m) * tan(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 <= 3.6d-88) then
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
    else
        tmp = ((l + l) / k_m) * ((l / (k_m * (sin(k_m) * tan(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 <= 3.6e-88) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l + l) / k_m) * ((l / (k_m * (Math.sin(k_m) * Math.tan(k_m)))) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.6e-88:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	else:
		tmp = ((l + l) / k_m) * ((l / (k_m * (math.sin(k_m) * math.tan(k_m)))) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.6e-88)
		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 + l) / k_m) * Float64(Float64(l / Float64(k_m * Float64(sin(k_m) * tan(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 <= 3.6e-88)
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	else
		tmp = ((l + l) / k_m) * ((l / (k_m * (sin(k_m) * tan(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, 3.6e-88], 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 + l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5999999999999999e-88

   1. Initial program 43.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 64.6

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

   5. Simplified 64.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 81.8

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

   7. Applied egg-rr 81.8%

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

   if 3.5999999999999999e-88 < k

   1. Initial program 23.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 76.0

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

   5. Simplified 76.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 80.4%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]

       2. div-inv N/A

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

       3. un-div-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot \tan k}} \cdot \frac{1}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       4. frac-times N/A

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

       5. *-rgt-identity N/A

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

       6. /-lowering-/.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. sin-lowering-sin.f64 N/A

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

       10. tan-lowering-tan.f64 98.4

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

   11. Applied egg-rr 98.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+180}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(k\_m \cdot t\right)} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 1e+180)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (* l (* (/ l (* (* (sin k_m) (tan k_m)) (* k_m t))) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 1e+180) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = l * ((l / ((sin(k_m) * tan(k_m)) * (k_m * t))) * 2.0);
	}
	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 ((l * l) <= 1d+180) then
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
    else
        tmp = l * ((l / ((sin(k_m) * tan(k_m)) * (k_m * t))) * 2.0d0)
    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 ((l * l) <= 1e+180) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = l * ((l / ((Math.sin(k_m) * Math.tan(k_m)) * (k_m * t))) * 2.0);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 1e+180:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	else:
		tmp = l * ((l / ((math.sin(k_m) * math.tan(k_m)) * (k_m * t))) * 2.0)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 1e+180)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(k_m * t))) * 2.0));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 1e+180)
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	else
		tmp = l * ((l / ((sin(k_m) * tan(k_m)) * (k_m * t))) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+180], 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[(l * N[(N[(l / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e180

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 67.5

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

   5. Simplified 67.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 85.9

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

   7. Applied egg-rr 85.9%

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

   if 1e180 < (*.f64 l l)

   1. Initial program 43.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 70.3

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

   5. Simplified 70.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 92.8%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]
   10. Step-by-step derivation

       1. flip-+ N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{\color{blue}{\frac{\ell \cdot \ell - \ell \cdot \ell}{\ell - \ell}}}{k} \]

       2. +-inverses N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{\frac{\color{blue}{0}}{\ell - \ell}}{k} \]

       3. +-inverses N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{\frac{0}{\color{blue}{0}}}{k} \]

       4. associate-/l/ N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \color{blue}{\frac{0}{k \cdot 0}} \]

       5. +-inverses N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{0}{k \cdot \color{blue}{\left(k - k\right)}} \]

       6. distribute-lft-out-- N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{0}{\color{blue}{k \cdot k - k \cdot k}} \]

       7. +-inverses N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{0}{\color{blue}{0}} \]

       8. +-inverses N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0} \]

       9. +-inverses N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t} \cdot \frac{\ell \cdot \ell - \ell \cdot \ell}{\color{blue}{\ell - \ell}} \]

       10. flip-+ N/A

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

       11. count-2 N/A

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

       12. associate-*r* N/A

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

       13. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 29.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+180}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot t\right)} \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell + \ell}{k\_m} \cdot \frac{\ell}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(k\_m \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (+ l l) k_m) (/ l (* (* (sin k_m) (tan k_m)) (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l + l) / k_m) * (l / ((sin(k_m) * tan(k_m)) * (k_m * t)));
}

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 = ((l + l) / k_m) * (l / ((sin(k_m) * tan(k_m)) * (k_m * t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l + l) / k_m) * (l / ((Math.sin(k_m) * Math.tan(k_m)) * (k_m * t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l + l) / k_m) * (l / ((math.sin(k_m) * math.tan(k_m)) * (k_m * t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l + l) / k_m) * Float64(l / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(k_m * t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l + l) / k_m) * (l / ((sin(k_m) * tan(k_m)) * (k_m * t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.6%

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

3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 77.2

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

5. Simplified 77.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

7. Applied egg-rr 83.9%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

9. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]
10. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k \cdot t}} \cdot \frac{\ell + \ell}{k} \]

    2. un-div-inv N/A

       \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot \tan k}}}{k \cdot t} \cdot \frac{\ell + \ell}{k} \]

    3. associate-/l/ N/A

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

    4. *-commutative N/A

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

    5. /-lowering-/.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. sin-lowering-sin.f64 N/A

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

    9. tan-lowering-tan.f64 N/A

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

    10. *-lowering-*.f64 94.3

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

11. Applied egg-rr 94.3%

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

12. Final simplification 94.3%

    \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot t\right)} \]
13. Add Preprocessing

Alternative 13: 76.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell + \ell}{k\_m}\\ \mathbf{if}\;k\_m \leq 1.85 \cdot 10^{+58}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\frac{\mathsf{fma}\left(\ell, \left(k\_m \cdot k\_m\right) \cdot -0.16666666666666666, \ell\right)}{k\_m \cdot k\_m}}{k\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (+ l l) k_m)))
   (if (<= k_m 1.85e+58)
     (*
      t_1
      (/
       (/ (/ (fma l (* (* k_m k_m) -0.16666666666666666) l) (* k_m k_m)) k_m)
       t))
     (* t_1 (/ l (* (* k_m t) (- 0.5 (* 0.5 (cos (+ k_m k_m))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (l + l) / k_m;
	double tmp;
	if (k_m <= 1.85e+58) {
		tmp = t_1 * (((fma(l, ((k_m * k_m) * -0.16666666666666666), l) / (k_m * k_m)) / k_m) / t);
	} else {
		tmp = t_1 * (l / ((k_m * t) * (0.5 - (0.5 * cos((k_m + k_m))))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(l + l) / k_m)
	tmp = 0.0
	if (k_m <= 1.85e+58)
		tmp = Float64(t_1 * Float64(Float64(Float64(fma(l, Float64(Float64(k_m * k_m) * -0.16666666666666666), l) / Float64(k_m * k_m)) / k_m) / t));
	else
		tmp = Float64(t_1 * Float64(l / Float64(Float64(k_m * t) * Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(l + l), $MachinePrecision] / k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 1.85e+58], N[(t$95$1 * N[(N[(N[(N[(l * N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8500000000000001e58

   1. Initial program 41.4%

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

   3. Taylor expanded in 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 79.3

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

   5. Simplified 79.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 82.2%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

   9. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \frac{1}{\sin k \cdot \tan k}}{k}}{t}} \cdot \frac{\ell + \ell}{k} \]

   10. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{\ell + \frac{-1}{6} \cdot \left({k}^{2} \cdot \ell\right)}{{k}^{2}}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-1}{6} \cdot \left({k}^{2} \cdot \ell\right) + \ell}}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\frac{-1}{6} \cdot {k}^{2}\right) \cdot \ell} + \ell}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \left(\frac{-1}{6} \cdot {k}^{2}\right)} + \ell}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot {k}^{2}, \ell\right)}}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \color{blue}{\frac{-1}{6} \cdot {k}^{2}}, \ell\right)}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot \color{blue}{\left(k \cdot k\right)}, \ell\right)}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot \color{blue}{\left(k \cdot k\right)}, \ell\right)}{{k}^{2}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, \frac{-1}{6} \cdot \left(k \cdot k\right), \ell\right)}{\color{blue}{k \cdot k}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

       10. *-lowering-*.f64 73.4

           \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\ell, -0.16666666666666666 \cdot \left(k \cdot k\right), \ell\right)}{\color{blue}{k \cdot k}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

   12. Simplified 73.4%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(\ell, -0.16666666666666666 \cdot \left(k \cdot k\right), \ell\right)}{k \cdot k}}}{k}}{t} \cdot \frac{\ell + \ell}{k} \]

   if 1.8500000000000001e58 < k

   1. Initial program 20.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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. /-lowering-/.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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. pow-lowering-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. sin-lowering-sin.f64 67.9

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

   5. Simplified 67.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell + \ell}{k}} \]

   8. Taylor expanded in k around 0

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

   10. Simplified 55.5%

       \[\leadsto \frac{\color{blue}{\ell}}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell + \ell}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+58}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\frac{\frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.16666666666666666, \ell\right)}{k \cdot k}}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{k\_m + k\_m}{k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.2e-106)
   (/ (+ k_m k_m) (* k_m (* t (* k_m (* k_m k_m)))))
   (/ (* (* l l) 2.0) (* k_m (* k_m (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-106) {
		tmp = (k_m + k_m) / (k_m * (t * (k_m * (k_m * k_m))));
	} else {
		tmp = ((l * l) * 2.0) / (k_m * (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 <= 7.2d-106) then
        tmp = (k_m + k_m) / (k_m * (t * (k_m * (k_m * k_m))))
    else
        tmp = ((l * l) * 2.0d0) / (k_m * (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 <= 7.2e-106) {
		tmp = (k_m + k_m) / (k_m * (t * (k_m * (k_m * k_m))));
	} else {
		tmp = ((l * l) * 2.0) / (k_m * (k_m * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.2e-106:
		tmp = (k_m + k_m) / (k_m * (t * (k_m * (k_m * k_m))))
	else:
		tmp = ((l * l) * 2.0) / (k_m * (k_m * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-106)
		tmp = Float64(Float64(k_m + k_m) / Float64(k_m * Float64(t * Float64(k_m * Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(l * l) * 2.0) / Float64(k_m * Float64(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 <= 7.2e-106)
		tmp = (k_m + k_m) / (k_m * (t * (k_m * (k_m * k_m))));
	else
		tmp = ((l * l) * 2.0) / (k_m * (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, 7.2e-106], N[(N[(k$95$m + k$95$m), $MachinePrecision] / N[(k$95$m * N[(t * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.20000000000000025e-106

   1. Initial program 42.9%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 64.6

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

   5. Simplified 64.6%

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

      1. flip-+ N/A

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

      2. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      3. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. +-inverses N/A

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\ell - \ell\right)}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      7. +-inverses N/A

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

      8. flip-+ N/A

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

      9. +-lowering-+.f64 44.5

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

   7. Applied egg-rr 44.5%

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

      1. /-lowering-/.f64 N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{\frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      4. +-inverses N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{\frac{k \cdot k - k \cdot k}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. cube-unmult N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 39.2

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

   9. Applied egg-rr 39.2%

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

   if 7.20000000000000025e-106 < k

   1. Initial program 25.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.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. *-lowering-*.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. +-lowering-+.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 59.3

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

   5. Simplified 59.3%

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

      1. flip-+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell \cdot \color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      4. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\ell \cdot 0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell \cdot \frac{0}{0}\right)}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      6. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      7. +-inverses N/A

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

      8. flip-+ N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 49.1

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

   7. Applied egg-rr 49.1%

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

      1. flip-+ N/A

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

      2. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \left(\frac{\color{blue}{0}}{\ell - \ell} \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      3. +-inverses N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \left(\frac{k \cdot k - k \cdot k}{\color{blue}{0}} \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. +-inverses N/A

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

      6. flip-+ N/A

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

      7. flip3-+ N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 33.6%

      \[\leadsto \frac{\ell \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(k, k, 0\right) \cdot \left(k + k\right)\right) \cdot \frac{1}{\mathsf{fma}\left(k, k, 0\right)}\right)} \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   10. Taylor expanded in l around 0

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. cube-mult N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-lowering-*.f64 53.9

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

   12. Simplified 53.9%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{k + k}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.7% accurate, 25.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.16:\\ \;\;\;\;\frac{4}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (if (<= k_m 0.16) (/ 4.0 0.0) 0.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.16) {
		tmp = 4.0 / 0.0;
	} 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 <= 0.16d0) then
        tmp = 4.0d0 / 0.0d0
    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 <= 0.16) {
		tmp = 4.0 / 0.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.16:
		tmp = 4.0 / 0.0
	else:
		tmp = 0.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.16)
		tmp = Float64(4.0 / 0.0);
	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 <= 0.16)
		tmp = 4.0 / 0.0;
	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, 0.16], N[(4.0 / 0.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 0.160000000000000003

   1. Initial program 42.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 53.1

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

   4. Applied egg-rr 53.1%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 44.9%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}} \]

      3. mul0-rgt N/A

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{0}} \]

      4. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{0}{2}}} \]

      5. mul0-rgt N/A

         \[\leadsto 2 \cdot \frac{1}{\frac{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}}{2}} \]

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. associate-*r/ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{4}}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{4}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}} \]

      12. mul0-rgt 30.1

          \[\leadsto \frac{4}{\color{blue}{0}} \]

   9. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{\frac{4}{0}} \]

   if 0.160000000000000003 < k

   1. Initial program 22.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. tan-lowering-tan.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. div-inv N/A

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

      14. *-lowering-*.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-lowering-*.f64 36.5

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

   4. Applied egg-rr 36.5%

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

   5. Taylor expanded in k around 0

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

   7. Simplified 10.7%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

      3. mul0-rgt N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\color{blue}{0}} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      6. metadata-eval N/A

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

      7. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

      10. mul0-rgt N/A

          \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{\color{blue}{0}} \]

      12. mul0-rgt N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

      13. metadata-eval N/A

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

   9. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\frac{-1}{0}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{0}{-1}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{0}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{\color{blue}{0 - 0}} \]

       4. flip-- N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0} - 0 \cdot 0}{0 + 0}} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{0 - \color{blue}{0}}{0 + 0}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0}}{0 + 0}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{0}{\color{blue}{0}}} \]

       9. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k - k}}{0}} \]

       10. +-inverses N/A

           \[\leadsto \frac{1}{\frac{k - k}{\color{blue}{k \cdot k - k \cdot k}}} \]

       11. clear-num N/A

           \[\leadsto \color{blue}{\frac{k \cdot k - k \cdot k}{k - k}} \]

       12. +-inverses N/A

           \[\leadsto \frac{\color{blue}{0}}{k - k} \]

       13. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{0 - 0}}{k - k} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{0 \cdot 0} - 0}{k - k} \]

       15. metadata-eval N/A

           \[\leadsto \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{k - k} \]

       16. +-inverses N/A

           \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

       17. metadata-eval N/A

           \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

       18. flip-- N/A

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

       19. metadata-eval 49.9

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

   11. Applied egg-rr 49.9%

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

Alternative 16: 28.8% accurate, 462.0× 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 37.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. cube-mult N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. tan-lowering-tan.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. div-inv N/A

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

   14. *-lowering-*.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. *-lowering-*.f64 49.4

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

4. Applied egg-rr 49.4%

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

5. Taylor expanded in k around 0

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

7. Simplified 37.3%

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

   1. clear-num N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}}{2}} \]

   3. mul0-rgt N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{0}}{2}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\color{blue}{0}} \]

   5. mul0-rgt N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

   6. metadata-eval N/A

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

   7. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(1 - 1\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{-1}{\mathsf{neg}\left(\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \color{blue}{0}\right)} \]

   10. mul0-rgt N/A

       \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{-1}{\color{blue}{0}} \]

   12. mul0-rgt N/A

       \[\leadsto \frac{-1}{\color{blue}{\left(\left(\left(\sin k \cdot \tan k\right) \cdot t\right) \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot 0}} \]

   13. metadata-eval N/A

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

9. Applied egg-rr 21.9%

   \[\leadsto \color{blue}{\frac{-1}{0}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{0}{-1}}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{1}{\color{blue}{0}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{1}{\color{blue}{0 - 0}} \]

    4. flip-- N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{0 \cdot 0 - 0 \cdot 0}{0 + 0}}} \]

    5. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{0} - 0 \cdot 0}{0 + 0}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{0 - \color{blue}{0}}{0 + 0}} \]

    7. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{0}}{0 + 0}} \]

    8. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{0}{\color{blue}{0}}} \]

    9. +-inverses N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{k - k}}{0}} \]

    10. +-inverses N/A

        \[\leadsto \frac{1}{\frac{k - k}{\color{blue}{k \cdot k - k \cdot k}}} \]

    11. clear-num N/A

        \[\leadsto \color{blue}{\frac{k \cdot k - k \cdot k}{k - k}} \]

    12. +-inverses N/A

        \[\leadsto \frac{\color{blue}{0}}{k - k} \]

    13. metadata-eval N/A

        \[\leadsto \frac{\color{blue}{0 - 0}}{k - k} \]

    14. metadata-eval N/A

        \[\leadsto \frac{\color{blue}{0 \cdot 0} - 0}{k - k} \]

    15. metadata-eval N/A

        \[\leadsto \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{k - k} \]

    16. +-inverses N/A

        \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

    17. metadata-eval N/A

        \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

    18. flip-- N/A

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

    19. metadata-eval 26.4

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

11. Applied egg-rr 26.4%

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

Reproduce

herbie shell --seed 2024198 
(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))))