Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 83.0%

Time: 24.2s

Alternatives: 9

Speedup: 2.0×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.3% 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: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\log \left({\left(e^{{k_m}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{{\ell}^{2}}{k_m} \cdot \frac{\cos k_m}{t \cdot {\sin k_m}^{2}}\right)}{k_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.35e-147)
   (/ 2.0 (* (log (pow (exp (pow k_m 4.0)) (pow l -2.0))) t))
   (/
    (* 2.0 (* (/ (pow l 2.0) k_m) (/ (cos k_m) (* t (pow (sin k_m) 2.0)))))
    k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.35e-147) {
		tmp = 2.0 / (log(pow(exp(pow(k_m, 4.0)), pow(l, -2.0))) * t);
	} else {
		tmp = (2.0 * ((pow(l, 2.0) / k_m) * (cos(k_m) / (t * pow(sin(k_m), 2.0))))) / k_m;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.35d-147) then
        tmp = 2.0d0 / (log((exp((k_m ** 4.0d0)) ** (l ** (-2.0d0)))) * t)
    else
        tmp = (2.0d0 * (((l ** 2.0d0) / k_m) * (cos(k_m) / (t * (sin(k_m) ** 2.0d0))))) / k_m
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.35e-147) {
		tmp = 2.0 / (Math.log(Math.pow(Math.exp(Math.pow(k_m, 4.0)), Math.pow(l, -2.0))) * t);
	} else {
		tmp = (2.0 * ((Math.pow(l, 2.0) / k_m) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0))))) / k_m;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.35e-147:
		tmp = 2.0 / (math.log(math.pow(math.exp(math.pow(k_m, 4.0)), math.pow(l, -2.0))) * t)
	else:
		tmp = (2.0 * ((math.pow(l, 2.0) / k_m) * (math.cos(k_m) / (t * math.pow(math.sin(k_m), 2.0))))) / k_m
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.35e-147)
		tmp = Float64(2.0 / Float64(log((exp((k_m ^ 4.0)) ^ (l ^ -2.0))) * t));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64((l ^ 2.0) / k_m) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0))))) / k_m);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.35e-147)
		tmp = 2.0 / (log((exp((k_m ^ 4.0)) ^ (l ^ -2.0))) * t);
	else
		tmp = (2.0 * (((l ^ 2.0) / k_m) * (cos(k_m) / (t * (sin(k_m) ^ 2.0))))) / k_m;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.35e-147], N[(2.0 / N[(N[Log[N[Power[N[Exp[N[Power[k$95$m, 4.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.34999999999999994e-147

   1. Initial program 43.0%

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

      1. associate-*l* 43.0%

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

      2. associate--l+ 43.0%

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

   3. Simplified 43.0%

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

      1. unpow3 43.1%

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

      2. times-frac 51.4%

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

      3. pow2 51.4%

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

   6. Applied egg-rr 51.4%

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

   7. Taylor expanded in k around 0 69.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 69.7%

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

   9. Simplified 69.7%

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

       1. div-inv 69.7%

          \[\leadsto \frac{2}{\color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot t} \]

       2. pow-flip 69.7%

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

       3. metadata-eval 69.7%

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

       4. add-log-exp 67.6%

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

       5. exp-prod 72.9%

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

   11. Applied egg-rr 72.9%

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

   if 2.34999999999999994e-147 < k

   1. Initial program 32.2%

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

      1. associate-/r* 32.1%

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

      2. associate-*l* 32.1%

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

      3. associate-*l/ 32.1%

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

      4. associate-/l* 32.1%

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

      5. +-commutative 32.1%

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

      6. unpow2 32.1%

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

      7. sqr-neg 32.1%

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

      8. distribute-frac-neg 32.1%

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

      9. distribute-frac-neg 32.1%

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

      10. unpow2 32.1%

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

      11. associate--l+ 41.1%

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

      12. metadata-eval 41.1%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 41.1%

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

      14. unpow2 41.1%

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

      15. distribute-frac-neg 41.1%

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

      16. distribute-frac-neg 41.1%

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

      17. sqr-neg 41.1%

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

      18. unpow2 41.1%

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

   3. Simplified 41.1%

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

      1. *-un-lft-identity 41.1%

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

      2. unpow2 41.1%

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

      3. times-frac 49.1%

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

      4. clear-num 49.1%

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

      5. associate-/r/ 49.1%

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

      6. associate-/r* 49.1%

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

      7. pow2 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. associate-*l/ 49.1%

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

      2. associate-*r/ 49.1%

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

      3. associate-*r/ 49.1%

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

      4. associate-/l/ 50.0%

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

      5. *-lft-identity 50.0%

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

      6. associate-*l/ 50.0%

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

      7. associate-*r/ 50.0%

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

      8. associate-*l/ 50.0%

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

      9. *-lft-identity 50.0%

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

   8. Simplified 50.0%

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

      1. associate-*l/ 50.9%

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

      2. associate-/l/ 50.9%

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

   10. Applied egg-rr 50.9%

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

   11. Taylor expanded in t around 0 82.9%

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

       1. *-commutative 82.9%

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

   13. Simplified 82.9%

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

       1. times-frac 83.9%

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

       2. *-commutative 83.9%

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

   15. Applied egg-rr 83.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {k_m}^{2}}{\cos k_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{k_m \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)\right)}}{k_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-5)
   (/ 2.0 (* (/ (pow k_m 2.0) (pow l 2.0)) (/ (* t (pow k_m 2.0)) (cos k_m))))
   (/
    (*
     2.0
     (/
      (* (pow l 2.0) (cos k_m))
      (* k_m (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))
    k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-5) {
		tmp = 2.0 / ((pow(k_m, 2.0) / pow(l, 2.0)) * ((t * pow(k_m, 2.0)) / cos(k_m)));
	} else {
		tmp = (2.0 * ((pow(l, 2.0) * cos(k_m)) / (k_m * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))))) / k_m;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9.2d-5) then
        tmp = 2.0d0 / (((k_m ** 2.0d0) / (l ** 2.0d0)) * ((t * (k_m ** 2.0d0)) / cos(k_m)))
    else
        tmp = (2.0d0 * (((l ** 2.0d0) * cos(k_m)) / (k_m * (t * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))) / k_m
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-5) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) / Math.pow(l, 2.0)) * ((t * Math.pow(k_m, 2.0)) / Math.cos(k_m)));
	} else {
		tmp = (2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (k_m * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))))) / k_m;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9.2e-5:
		tmp = 2.0 / ((math.pow(k_m, 2.0) / math.pow(l, 2.0)) * ((t * math.pow(k_m, 2.0)) / math.cos(k_m)))
	else:
		tmp = (2.0 * ((math.pow(l, 2.0) * math.cos(k_m)) / (k_m * (t * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))) / k_m
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-5)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) / (l ^ 2.0)) * Float64(Float64(t * (k_m ^ 2.0)) / cos(k_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(k_m * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))))) / k_m);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9.2e-5)
		tmp = 2.0 / (((k_m ^ 2.0) / (l ^ 2.0)) * ((t * (k_m ^ 2.0)) / cos(k_m)));
	else
		tmp = (2.0 * (((l ^ 2.0) * cos(k_m)) / (k_m * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))))) / k_m;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-5], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.20000000000000001e-5

   1. Initial program 41.1%

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

      1. associate-*l* 41.1%

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

      2. associate--l+ 41.1%

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

   3. Simplified 41.1%

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

      1. unpow3 41.1%

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

      2. times-frac 49.5%

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

      3. pow2 49.5%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in t around 0 75.4%

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

      1. times-frac 78.1%

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

      2. *-commutative 78.1%

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

   9. Simplified 78.1%

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

   10. Taylor expanded in k around 0 75.5%

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

   if 9.20000000000000001e-5 < k

   1. Initial program 32.3%

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

      1. associate-/r* 32.1%

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

      2. associate-*l* 32.1%

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

      3. associate-*l/ 32.1%

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

      4. associate-/l* 32.1%

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

      5. +-commutative 32.1%

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

      6. unpow2 32.1%

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

      7. sqr-neg 32.1%

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

      8. distribute-frac-neg 32.1%

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

      9. distribute-frac-neg 32.1%

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

      10. unpow2 32.1%

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

      11. associate--l+ 37.4%

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

      12. metadata-eval 37.4%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 37.4%

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

      14. unpow2 37.4%

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

      15. distribute-frac-neg 37.4%

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

      16. distribute-frac-neg 37.4%

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

      17. sqr-neg 37.4%

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

      18. unpow2 37.4%

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

   3. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. unpow2 37.4%

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

      3. times-frac 47.4%

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

      4. clear-num 47.4%

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

      5. associate-/r/ 47.4%

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

      6. associate-/r* 47.4%

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

      7. pow2 47.4%

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

   6. Applied egg-rr 47.4%

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

      1. associate-*l/ 47.4%

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

      2. associate-*r/ 47.4%

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

      3. associate-*r/ 47.4%

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

      4. associate-/l/ 48.6%

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

      5. *-lft-identity 48.6%

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

      6. associate-*l/ 48.6%

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

      7. associate-*r/ 48.7%

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

      8. associate-*l/ 48.7%

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

      9. *-lft-identity 48.7%

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

   8. Simplified 48.7%

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

      1. associate-*l/ 48.7%

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

      2. associate-/l/ 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in t around 0 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

       1. unpow2 81.3%

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

       2. sin-mult 81.3%

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

   15. Applied egg-rr 81.3%

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

       1. div-sub 81.3%

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

       2. +-inverses 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right) \cdot t\right)}}{k} \]

       3. cos-0 81.3%

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

       4. metadata-eval 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right) \cdot t\right)}}{k} \]

       5. count-2 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right) \cdot t\right)}}{k} \]

       6. *-commutative 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right) \cdot t\right)}}{k} \]

   17. Simplified 81.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \left(\frac{{\ell}^{2}}{k_m} \cdot \frac{\cos k_m}{t \cdot {\sin k_m}^{2}}\right)}{k_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  (* 2.0 (* (/ (pow l 2.0) k_m) (/ (cos k_m) (* t (pow (sin k_m) 2.0)))))
  k_m))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * ((pow(l, 2.0) / k_m) * (cos(k_m) / (t * pow(sin(k_m), 2.0))))) / 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 = (2.0d0 * (((l ** 2.0d0) / k_m) * (cos(k_m) / (t * (sin(k_m) ** 2.0d0))))) / k_m
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * ((Math.pow(l, 2.0) / k_m) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0))))) / k_m;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * ((math.pow(l, 2.0) / k_m) * (math.cos(k_m) / (t * math.pow(math.sin(k_m), 2.0))))) / k_m

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * Float64(Float64((l ^ 2.0) / k_m) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0))))) / k_m)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * (((l ^ 2.0) / k_m) * (cos(k_m) / (t * (sin(k_m) ^ 2.0))))) / k_m;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-/r* 38.4%

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

   2. associate-*l* 38.4%

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

   3. associate-*l/ 38.4%

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

   4. associate-/l* 38.4%

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

   5. +-commutative 38.4%

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

   6. unpow2 38.4%

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

   7. sqr-neg 38.4%

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

   8. distribute-frac-neg 38.4%

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

   9. distribute-frac-neg 38.4%

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

   10. unpow2 38.4%

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

   11. associate--l+ 45.4%

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

   12. metadata-eval 45.4%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 45.4%

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

   14. unpow2 45.4%

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

   15. distribute-frac-neg 45.4%

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

   16. distribute-frac-neg 45.4%

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

   17. sqr-neg 45.4%

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

   18. unpow2 45.4%

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

3. Simplified 45.4%

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

   1. *-un-lft-identity 45.4%

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

   2. unpow2 45.4%

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

   3. times-frac 51.2%

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

   4. clear-num 51.2%

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

   5. associate-/r/ 51.2%

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

   6. associate-/r* 51.2%

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

   7. pow2 51.2%

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

6. Applied egg-rr 51.2%

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

   1. associate-*l/ 51.2%

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

   2. associate-*r/ 51.2%

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

   3. associate-*r/ 51.2%

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

   4. associate-/l/ 51.6%

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

   5. *-lft-identity 51.6%

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

   6. associate-*l/ 51.6%

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

   7. associate-*r/ 51.6%

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

   8. associate-*l/ 51.6%

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

   9. *-lft-identity 51.6%

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

8. Simplified 51.6%

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

   1. associate-*l/ 52.0%

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

   2. associate-/l/ 52.0%

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

10. Applied egg-rr 52.0%

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

11. Taylor expanded in t around 0 81.6%

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

    1. *-commutative 81.6%

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

13. Simplified 81.6%

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

    1. times-frac 82.1%

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

    2. *-commutative 82.1%

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

15. Applied egg-rr 82.1%

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

16. Final simplification 82.1%

    \[\leadsto \frac{2 \cdot \left(\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)}{k} \]
17. Add Preprocessing

Alternative 4: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\ell}^{2} \cdot \cos k_m\\ \mathbf{if}\;k_m \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{t_1}{k_m \cdot \left(t \cdot {k_m}^{2}\right)}}{k_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{t_1}{k_m \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)\right)}}{k_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (pow l 2.0) (cos k_m))))
   (if (<= k_m 9.2e-5)
     (/ (* 2.0 (/ t_1 (* k_m (* t (pow k_m 2.0))))) k_m)
     (/ (* 2.0 (/ t_1 (* k_m (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(l, 2.0) * cos(k_m);
	double tmp;
	if (k_m <= 9.2e-5) {
		tmp = (2.0 * (t_1 / (k_m * (t * pow(k_m, 2.0))))) / k_m;
	} else {
		tmp = (2.0 * (t_1 / (k_m * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))))) / 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) :: t_1
    real(8) :: tmp
    t_1 = (l ** 2.0d0) * cos(k_m)
    if (k_m <= 9.2d-5) then
        tmp = (2.0d0 * (t_1 / (k_m * (t * (k_m ** 2.0d0))))) / k_m
    else
        tmp = (2.0d0 * (t_1 / (k_m * (t * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))))) / 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 t_1 = Math.pow(l, 2.0) * Math.cos(k_m);
	double tmp;
	if (k_m <= 9.2e-5) {
		tmp = (2.0 * (t_1 / (k_m * (t * Math.pow(k_m, 2.0))))) / k_m;
	} else {
		tmp = (2.0 * (t_1 / (k_m * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))))) / k_m;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(l, 2.0) * math.cos(k_m)
	tmp = 0
	if k_m <= 9.2e-5:
		tmp = (2.0 * (t_1 / (k_m * (t * math.pow(k_m, 2.0))))) / k_m
	else:
		tmp = (2.0 * (t_1 / (k_m * (t * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))))) / k_m
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64((l ^ 2.0) * cos(k_m))
	tmp = 0.0
	if (k_m <= 9.2e-5)
		tmp = Float64(Float64(2.0 * Float64(t_1 / Float64(k_m * Float64(t * (k_m ^ 2.0))))) / k_m);
	else
		tmp = Float64(Float64(2.0 * Float64(t_1 / Float64(k_m * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))))) / k_m);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (l ^ 2.0) * cos(k_m);
	tmp = 0.0;
	if (k_m <= 9.2e-5)
		tmp = (2.0 * (t_1 / (k_m * (t * (k_m ^ 2.0))))) / k_m;
	else
		tmp = (2.0 * (t_1 / (k_m * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))))) / k_m;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 9.2e-5], N[(N[(2.0 * N[(t$95$1 / N[(k$95$m * N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], N[(N[(2.0 * N[(t$95$1 / N[(k$95$m * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.20000000000000001e-5

   1. Initial program 41.1%

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

      1. associate-/r* 41.1%

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

      2. associate-*l* 41.1%

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

      3. associate-*l/ 41.1%

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

      4. associate-/l* 41.1%

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

      5. +-commutative 41.1%

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

      6. unpow2 41.1%

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

      7. sqr-neg 41.1%

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

      8. distribute-frac-neg 41.1%

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

      9. distribute-frac-neg 41.1%

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

      10. unpow2 41.1%

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

      11. associate--l+ 48.8%

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

      12. metadata-eval 48.8%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 48.8%

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

      14. unpow2 48.8%

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

      15. distribute-frac-neg 48.8%

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

      16. distribute-frac-neg 48.8%

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

      17. sqr-neg 48.8%

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

      18. unpow2 48.8%

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

   3. Simplified 48.8%

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

      1. *-un-lft-identity 48.8%

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

      2. unpow2 48.8%

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

      3. times-frac 52.8%

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

      4. clear-num 52.8%

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

      5. associate-/r/ 52.8%

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

      6. associate-/r* 52.8%

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

      7. pow2 52.8%

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

   6. Applied egg-rr 52.8%

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

      1. associate-*l/ 52.9%

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

      2. associate-*r/ 52.9%

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

      3. associate-*r/ 52.9%

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

      4. associate-/l/ 52.9%

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

      5. *-lft-identity 52.9%

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

      6. associate-*l/ 52.9%

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

      7. associate-*r/ 52.9%

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

      8. associate-*l/ 52.9%

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

      9. *-lft-identity 52.9%

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

   8. Simplified 52.9%

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

      1. associate-*l/ 53.5%

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

      2. associate-/l/ 53.5%

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

   10. Applied egg-rr 53.5%

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

   11. Taylor expanded in t around 0 81.8%

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

       1. *-commutative 81.8%

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

   13. Simplified 81.8%

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

   14. Taylor expanded in k around 0 75.4%

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

   if 9.20000000000000001e-5 < k

   1. Initial program 32.3%

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

      1. associate-/r* 32.1%

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

      2. associate-*l* 32.1%

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

      3. associate-*l/ 32.1%

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

      4. associate-/l* 32.1%

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

      5. +-commutative 32.1%

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

      6. unpow2 32.1%

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

      7. sqr-neg 32.1%

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

      8. distribute-frac-neg 32.1%

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

      9. distribute-frac-neg 32.1%

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

      10. unpow2 32.1%

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

      11. associate--l+ 37.4%

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

      12. metadata-eval 37.4%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 37.4%

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

      14. unpow2 37.4%

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

      15. distribute-frac-neg 37.4%

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

      16. distribute-frac-neg 37.4%

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

      17. sqr-neg 37.4%

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

      18. unpow2 37.4%

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

   3. Simplified 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. unpow2 37.4%

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

      3. times-frac 47.4%

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

      4. clear-num 47.4%

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

      5. associate-/r/ 47.4%

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

      6. associate-/r* 47.4%

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

      7. pow2 47.4%

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

   6. Applied egg-rr 47.4%

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

      1. associate-*l/ 47.4%

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

      2. associate-*r/ 47.4%

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

      3. associate-*r/ 47.4%

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

      4. associate-/l/ 48.6%

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

      5. *-lft-identity 48.6%

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

      6. associate-*l/ 48.6%

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

      7. associate-*r/ 48.7%

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

      8. associate-*l/ 48.7%

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

      9. *-lft-identity 48.7%

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

   8. Simplified 48.7%

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

      1. associate-*l/ 48.7%

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

      2. associate-/l/ 48.7%

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

   10. Applied egg-rr 48.7%

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

   11. Taylor expanded in t around 0 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

       1. unpow2 81.3%

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

       2. sin-mult 81.3%

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

   15. Applied egg-rr 81.3%

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

       1. div-sub 81.3%

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

       2. +-inverses 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right) \cdot t\right)}}{k} \]

       3. cos-0 81.3%

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

       4. metadata-eval 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right) \cdot t\right)}}{k} \]

       5. count-2 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right) \cdot t\right)}}{k} \]

       6. *-commutative 81.3%

          \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right) \cdot t\right)}}{k} \]

   17. Simplified 81.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(t \cdot {k}^{2}\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k_m}{k_m \cdot \left(t \cdot {k_m}^{2}\right)}}{k_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* 2.0 (/ (* (pow l 2.0) (cos k_m)) (* k_m (* t (pow k_m 2.0))))) k_m))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * ((pow(l, 2.0) * cos(k_m)) / (k_m * (t * pow(k_m, 2.0))))) / 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 = (2.0d0 * (((l ** 2.0d0) * cos(k_m)) / (k_m * (t * (k_m ** 2.0d0))))) / k_m
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (k_m * (t * Math.pow(k_m, 2.0))))) / k_m;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * ((math.pow(l, 2.0) * math.cos(k_m)) / (k_m * (t * math.pow(k_m, 2.0))))) / k_m

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(k_m * Float64(t * (k_m ^ 2.0))))) / k_m)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * (((l ^ 2.0) * cos(k_m)) / (k_m * (t * (k_m ^ 2.0))))) / k_m;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-/r* 38.4%

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

   2. associate-*l* 38.4%

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

   3. associate-*l/ 38.4%

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

   4. associate-/l* 38.4%

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

   5. +-commutative 38.4%

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

   6. unpow2 38.4%

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

   7. sqr-neg 38.4%

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

   8. distribute-frac-neg 38.4%

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

   9. distribute-frac-neg 38.4%

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

   10. unpow2 38.4%

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

   11. associate--l+ 45.4%

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

   12. metadata-eval 45.4%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 45.4%

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

   14. unpow2 45.4%

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

   15. distribute-frac-neg 45.4%

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

   16. distribute-frac-neg 45.4%

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

   17. sqr-neg 45.4%

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

   18. unpow2 45.4%

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

3. Simplified 45.4%

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

   1. *-un-lft-identity 45.4%

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

   2. unpow2 45.4%

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

   3. times-frac 51.2%

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

   4. clear-num 51.2%

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

   5. associate-/r/ 51.2%

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

   6. associate-/r* 51.2%

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

   7. pow2 51.2%

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

6. Applied egg-rr 51.2%

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

   1. associate-*l/ 51.2%

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

   2. associate-*r/ 51.2%

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

   3. associate-*r/ 51.2%

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

   4. associate-/l/ 51.6%

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

   5. *-lft-identity 51.6%

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

   6. associate-*l/ 51.6%

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

   7. associate-*r/ 51.6%

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

   8. associate-*l/ 51.6%

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

   9. *-lft-identity 51.6%

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

8. Simplified 51.6%

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

   1. associate-*l/ 52.0%

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

   2. associate-/l/ 52.0%

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

10. Applied egg-rr 52.0%

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

11. Taylor expanded in t around 0 81.6%

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

    1. *-commutative 81.6%

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

13. Simplified 81.6%

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

14. Taylor expanded in k around 0 69.9%

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

15. Final simplification 69.9%

    \[\leadsto \frac{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(t \cdot {k}^{2}\right)}}{k} \]
16. Add Preprocessing

Alternative 6: 62.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \frac{{\ell}^{2}}{t \cdot {k_m}^{3}}}{k_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* 2.0 (/ (pow l 2.0) (* t (pow k_m 3.0)))) k_m))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * (pow(l, 2.0) / (t * pow(k_m, 3.0)))) / 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 = (2.0d0 * ((l ** 2.0d0) / (t * (k_m ** 3.0d0)))) / k_m
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 3.0)))) / k_m;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * (math.pow(l, 2.0) / (t * math.pow(k_m, 3.0)))) / k_m

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k_m ^ 3.0)))) / k_m)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * ((l ^ 2.0) / (t * (k_m ^ 3.0)))) / k_m;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-/r* 38.4%

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

   2. associate-*l* 38.4%

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

   3. associate-*l/ 38.4%

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

   4. associate-/l* 38.4%

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

   5. +-commutative 38.4%

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

   6. unpow2 38.4%

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

   7. sqr-neg 38.4%

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

   8. distribute-frac-neg 38.4%

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

   9. distribute-frac-neg 38.4%

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

   10. unpow2 38.4%

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

   11. associate--l+ 45.4%

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

   12. metadata-eval 45.4%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 45.4%

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

   14. unpow2 45.4%

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

   15. distribute-frac-neg 45.4%

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

   16. distribute-frac-neg 45.4%

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

   17. sqr-neg 45.4%

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

   18. unpow2 45.4%

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

3. Simplified 45.4%

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

   1. *-un-lft-identity 45.4%

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

   2. unpow2 45.4%

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

   3. times-frac 51.2%

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

   4. clear-num 51.2%

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

   5. associate-/r/ 51.2%

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

   6. associate-/r* 51.2%

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

   7. pow2 51.2%

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

6. Applied egg-rr 51.2%

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

   1. associate-*l/ 51.2%

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

   2. associate-*r/ 51.2%

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

   3. associate-*r/ 51.2%

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

   4. associate-/l/ 51.6%

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

   5. *-lft-identity 51.6%

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

   6. associate-*l/ 51.6%

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

   7. associate-*r/ 51.6%

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

   8. associate-*l/ 51.6%

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

   9. *-lft-identity 51.6%

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

8. Simplified 51.6%

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

   1. associate-*l/ 52.0%

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

   2. associate-/l/ 52.0%

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

10. Applied egg-rr 52.0%

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

11. Taylor expanded in k around 0 67.1%

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

12. Final simplification 67.1%

    \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{3}}}{k} \]
13. Add Preprocessing

Alternative 7: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{t \cdot \left({k_m}^{4} \cdot {\ell}^{-2}\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (* t (* (pow k_m 4.0) (pow l -2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (t * (pow(k_m, 4.0) * pow(l, -2.0)));
}

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 = 2.0d0 / (t * ((k_m ** 4.0d0) * (l ** (-2.0d0))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / (t * (Math.pow(k_m, 4.0) * Math.pow(l, -2.0)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (t * (math.pow(k_m, 4.0) * math.pow(l, -2.0)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(t * Float64((k_m ^ 4.0) * (l ^ -2.0))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / (t * ((k_m ^ 4.0) * (l ^ -2.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(t * N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate--l+ 38.4%

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

3. Simplified 38.4%

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

   1. unpow3 38.4%

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

   2. times-frac 45.2%

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

   3. pow2 45.2%

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

6. Applied egg-rr 45.2%

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

7. Taylor expanded in k around 0 64.9%

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

   1. associate-/l* 63.2%

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

   2. associate-/r/ 64.5%

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

9. Simplified 64.5%

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

    1. expm1-log1p-u 64.5%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{4}}{{\ell}^{2}}\right)\right)} \cdot t} \]

    2. expm1-udef 62.2%

       \[\leadsto \frac{2}{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{k}^{4}}{{\ell}^{2}}\right)} - 1\right)} \cdot t} \]

    3. div-inv 62.2%

       \[\leadsto \frac{2}{\left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{4} \cdot \frac{1}{{\ell}^{2}}}\right)} - 1\right) \cdot t} \]

    4. pow-flip 62.2%

       \[\leadsto \frac{2}{\left(e^{\mathsf{log1p}\left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)} - 1\right) \cdot t} \]

    5. metadata-eval 62.2%

       \[\leadsto \frac{2}{\left(e^{\mathsf{log1p}\left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)} - 1\right) \cdot t} \]

11. Applied egg-rr 62.2%

    \[\leadsto \frac{2}{\color{blue}{\left(e^{\mathsf{log1p}\left({k}^{4} \cdot {\ell}^{-2}\right)} - 1\right)} \cdot t} \]
12. Step-by-step derivation

    1. expm1-def 64.5%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{4} \cdot {\ell}^{-2}\right)\right)} \cdot t} \]

    2. expm1-log1p 64.5%

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

13. Simplified 64.5%

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

14. Final simplification 64.5%

    \[\leadsto \frac{2}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)} \]
15. Add Preprocessing

Alternative 8: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{{\ell}^{-2} \cdot \left({k_m}^{4} \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (* (pow l -2.0) (* (pow k_m 4.0) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (pow(l, -2.0) * (pow(k_m, 4.0) * 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 = 2.0d0 / ((l ** (-2.0d0)) * ((k_m ** 4.0d0) * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / (Math.pow(l, -2.0) * (Math.pow(k_m, 4.0) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (math.pow(l, -2.0) * (math.pow(k_m, 4.0) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64((l ^ -2.0) * Float64((k_m ^ 4.0) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((l ^ -2.0) * ((k_m ^ 4.0) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(N[Power[k$95$m, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate--l+ 38.4%

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

3. Simplified 38.4%

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

5. Taylor expanded in k around 0 64.9%

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

   1. expm1-log1p-u 44.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)\right)}} \]

   2. expm1-udef 23.0%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)} - 1}} \]

   3. div-inv 23.0%

      \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left({k}^{4} \cdot t\right) \cdot \frac{1}{{\ell}^{2}}}\right)} - 1} \]

   4. *-commutative 23.0%

      \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left(t \cdot {k}^{4}\right)} \cdot \frac{1}{{\ell}^{2}}\right)} - 1} \]

   5. pow-flip 23.0%

      \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)} - 1} \]

   6. metadata-eval 23.0%

      \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{\color{blue}{-2}}\right)} - 1} \]

7. Applied egg-rr 23.0%

   \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 44.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}\right)\right)}} \]

   2. expm1-log1p 64.9%

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

   3. *-commutative 64.9%

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

   4. *-commutative 64.9%

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

9. Simplified 64.9%

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

10. Final simplification 64.9%

    \[\leadsto \frac{2}{{\ell}^{-2} \cdot \left({k}^{4} \cdot t\right)} \]
11. Add Preprocessing

Alternative 9: 61.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{{k_m}^{4} \cdot t}{{\ell}^{2}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (/ (* (pow k_m 4.0) t) (pow l 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((pow(k_m, 4.0) * t) / pow(l, 2.0));
}

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 = 2.0d0 / (((k_m ** 4.0d0) * t) / (l ** 2.0d0))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / ((Math.pow(k_m, 4.0) * t) / Math.pow(l, 2.0));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / ((math.pow(k_m, 4.0) * t) / math.pow(l, 2.0))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64((k_m ^ 4.0) * t) / (l ^ 2.0)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / (((k_m ^ 4.0) * t) / (l ^ 2.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] * t), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. associate-*l* 38.4%

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

   2. associate--l+ 38.4%

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

3. Simplified 38.4%

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

5. Taylor expanded in k around 0 64.9%

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

6. Final simplification 64.9%

   \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))))