Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 93.4%

Time: 16.0s

Alternatives: 10

Speedup: 11.0×

• 40.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{t \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)}}{k\_m} \cdot \frac{\ell \cdot \cos k\_m}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0001)
   (* (/ (* 2.0 l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (*
    (/ (/ (* 2.0 l) (* t (fma (cos (+ k_m k_m)) -0.5 0.5))) k_m)
    (/ (* l (cos k_m)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0001) {
		tmp = ((2.0 * l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = (((2.0 * l) / (t * fma(cos((k_m + k_m)), -0.5, 0.5))) / k_m) * ((l * cos(k_m)) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0001)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / Float64(t * fma(cos(Float64(k_m + k_m)), -0.5, 0.5))) / k_m) * Float64(Float64(l * cos(k_m)) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0001], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000005e-4

   1. Initial program 42.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   7. Applied rewrites 82.7%

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

   if 1.00000000000000005e-4 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 96.0%

      \[\leadsto \frac{2 \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{t \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k} \cdot \frac{\ell \cdot \cos k}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{2 \cdot \ell}{k\_m \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0001)
   (* (/ (* 2.0 l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (*
    (/ (* l (cos k_m)) k_m)
    (/ (* 2.0 l) (* k_m (* t (fma (cos (+ k_m k_m)) -0.5 0.5)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0001) {
		tmp = ((2.0 * l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = ((l * cos(k_m)) / k_m) * ((2.0 * l) / (k_m * (t * fma(cos((k_m + k_m)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0001)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(2.0 * l) / Float64(k_m * Float64(t * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0001], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * N[(t * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000005e-4

   1. Initial program 42.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   7. Applied rewrites 82.7%

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

   if 1.00000000000000005e-4 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 96.0%

      \[\leadsto \frac{2 \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{2 \cdot \ell}{k \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \frac{\cos k\_m}{k\_m}}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0001)
   (* (/ (* 2.0 l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (*
    (* 2.0 l)
    (/
     (* l (/ (cos k_m) k_m))
     (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0001) {
		tmp = ((2.0 * l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = (2.0 * l) * ((l * (cos(k_m) / k_m)) / (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0001)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(l * Float64(cos(k_m) / k_m)) / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0001], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000005e-4

   1. Initial program 42.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   7. Applied rewrites 82.7%

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

   if 1.00000000000000005e-4 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 96.0%

      \[\leadsto \frac{2 \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 94.7%

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

Alternative 4: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k\_m}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0001)
   (* (/ (* 2.0 l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (*
    (* 2.0 l)
    (/
     (* l (cos k_m))
     (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0001) {
		tmp = ((2.0 * l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = (2.0 * l) * ((l * cos(k_m)) / (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * (k_m * t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0001)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(l * cos(k_m)) / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * Float64(k_m * t)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0001], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000005e-4

   1. Initial program 42.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   7. Applied rewrites 82.7%

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

   if 1.00000000000000005e-4 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 96.0%

      \[\leadsto \frac{2 \cdot \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 85.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0001:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{elif}\;k\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{k\_m \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2 \cdot \ell}{k\_m \cdot k\_m}}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-87)
   (* (/ (* 2.0 l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (if (<= k_m 5.6e+102)
     (*
      (/ (* l (cos k_m)) k_m)
      (/
       (* (/ l t) (fma 0.6666666666666666 (* k_m k_m) 2.0))
       (* k_m (* k_m k_m))))
     (* (/ l t) (/ (/ (* 2.0 l) (* k_m k_m)) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-87) {
		tmp = ((2.0 * l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else if (k_m <= 5.6e+102) {
		tmp = ((l * cos(k_m)) / k_m) * (((l / t) * fma(0.6666666666666666, (k_m * k_m), 2.0)) / (k_m * (k_m * k_m)));
	} else {
		tmp = (l / t) * (((2.0 * l) / (k_m * k_m)) / (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-87)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	elseif (k_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(Float64(l / t) * fma(0.6666666666666666, Float64(k_m * k_m), 2.0)) / Float64(k_m * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(l / t) * Float64(Float64(Float64(2.0 * l) / Float64(k_m * k_m)) / Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-87], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.6e+102], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.19999999999999979e-87

   1. Initial program 42.3%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 80.2%

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

   if 3.19999999999999979e-87 < k < 5.60000000000000037e102

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 81.4

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.4%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{k} \]

   if 5.60000000000000037e102 < k

   1. Initial program 30.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 54.4%

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

   9. Applied rewrites 58.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2 \cdot \ell}{k \cdot k}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.6

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

5. Applied rewrites 62.6%

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

7. Applied rewrites 74.2%

   \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]
8. Add Preprocessing

Alternative 7: 71.0% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 62.6

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

5. Applied rewrites 62.6%

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

7. Applied rewrites 71.1%

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

8. Final simplification 71.1%

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

Alternative 8: 20.2% accurate, 14.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{1}{\frac{t}{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 1.0 (/ t (* -0.11666666666666667 (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 1.0 / (t / (-0.11666666666666667 * (l * l)));
}

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 = 1.0d0 / (t / ((-0.11666666666666667d0) * (l * l)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 1.0 / (t / (-0.11666666666666667 * (l * l)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 1.0 / (t / (-0.11666666666666667 * (l * l)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(1.0 / Float64(t / Float64(-0.11666666666666667 * Float64(l * l))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 1.0 / (t / (-0.11666666666666667 * (l * l)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(1.0 / N[(t / N[(-0.11666666666666667 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Applied rewrites 24.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 16.5%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{\color{blue}{t}} \]
9. Step-by-step derivation

10. Applied rewrites 16.5%

    \[\leadsto \frac{1}{\frac{t}{\color{blue}{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}}} \]

11. Final simplification 16.5%

    \[\leadsto \frac{1}{\frac{t}{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}} \]
12. Add Preprocessing

Alternative 9: 20.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) * (-0.11666666666666667 / t);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) * ((-0.11666666666666667d0) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * l) * (-0.11666666666666667 / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Applied rewrites 24.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 16.5%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{\color{blue}{t}} \]
9. Step-by-step derivation

10. Applied rewrites 16.5%

    \[\leadsto \frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \color{blue}{\ell}\right) \]

11. Final simplification 16.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
12. Add Preprocessing

Alternative 10: 17.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\ell \cdot \frac{-0.11666666666666667}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* l (* l (/ -0.11666666666666667 t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l * (-0.11666666666666667 / t));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l * ((-0.11666666666666667d0) / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l * (-0.11666666666666667 / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l * (-0.11666666666666667 / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l * Float64(-0.11666666666666667 / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l * (-0.11666666666666667 / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Applied rewrites 24.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 16.5%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{\color{blue}{t}} \]
9. Step-by-step derivation

10. Applied rewrites 13.3%

    \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{-0.11666666666666667}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))))