Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.4% → 95.0%

Time: 14.6s

Alternatives: 18

Speedup: 11.6×

• 40.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.4% 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: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell \cdot \cos k\_m}\\ \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k\_m}^{2} \cdot \left(\frac{t}{\ell} \cdot k\_m\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* l (cos k_m)))))
   (if (<= k_m 1.8e-112)
     (/
      2.0
      (*
       (*
        k_m
        (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
       t_1))
     (/ 2.0 (* (* (pow (sin k_m) 2.0) (* (/ t l) k_m)) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (l * cos(k_m));
	double tmp;
	if (k_m <= 1.8e-112) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * t_1);
	} else {
		tmp = 2.0 / ((pow(sin(k_m), 2.0) * ((t / l) * k_m)) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(l * cos(k_m)))
	tmp = 0.0
	if (k_m <= 1.8e-112)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k_m) ^ 2.0) * Float64(Float64(t / l) * k_m)) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.8e-112], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-112

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.5%

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

   if 1.8e-112 < k

   1. Initial program 24.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 96.5%

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

Alternative 2: 89.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot 2\\ \mathbf{if}\;k\_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \mathbf{elif}\;k\_m \leq 6.4 \cdot 10^{+99}:\\ \;\;\;\;\ell \cdot \left(t\_1 \cdot \frac{\ell}{t \cdot {\left(\sin k\_m \cdot k\_m\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell \cdot \ell}{k\_m}}{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) 2.0)))
   (if (<= k_m 1.4e-8)
     (/
      2.0
      (*
       (*
        k_m
        (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
       (/ k_m (* l (cos k_m)))))
     (if (<= k_m 6.4e+99)
       (* l (* t_1 (/ l (* t (pow (* (sin k_m) k_m) 2.0)))))
       (* t_1 (/ (/ (* l l) k_m) (* (* (pow (sin k_m) 2.0) t) k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * 2.0;
	double tmp;
	if (k_m <= 1.4e-8) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * (k_m / (l * cos(k_m))));
	} else if (k_m <= 6.4e+99) {
		tmp = l * (t_1 * (l / (t * pow((sin(k_m) * k_m), 2.0))));
	} else {
		tmp = t_1 * (((l * l) / k_m) / ((pow(sin(k_m), 2.0) * t) * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * 2.0)
	tmp = 0.0
	if (k_m <= 1.4e-8)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * Float64(k_m / Float64(l * cos(k_m)))));
	elseif (k_m <= 6.4e+99)
		tmp = Float64(l * Float64(t_1 * Float64(l / Float64(t * (Float64(sin(k_m) * k_m) ^ 2.0)))));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l * l) / k_m) / Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[k$95$m, 1.4e-8], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6.4e+99], N[(l * N[(t$95$1 * N[(l / N[(t * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4e-8

   1. Initial program 37.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 83.1%

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

   if 1.4e-8 < k < 6.39999999999999999e99

   1. Initial program 7.1%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 82.4%

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

   9. Applied rewrites 92.7%

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

   if 6.39999999999999999e99 < k

   1. Initial program 18.2%

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

   3. Taylor expanded in 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. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 71.3%

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

Alternative 3: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell \cdot \cos k\_m}\\ \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k\_m}^{2} \cdot \frac{t}{\ell}\right) \cdot k\_m\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* l (cos k_m)))))
   (if (<= k_m 1.8e-112)
     (/
      2.0
      (*
       (*
        k_m
        (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
       t_1))
     (/ 2.0 (* (* (* (pow (sin k_m) 2.0) (/ t l)) k_m) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (l * cos(k_m));
	double tmp;
	if (k_m <= 1.8e-112) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * t_1);
	} else {
		tmp = 2.0 / (((pow(sin(k_m), 2.0) * (t / l)) * k_m) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(l * cos(k_m)))
	tmp = 0.0
	if (k_m <= 1.8e-112)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k_m) ^ 2.0) * Float64(t / l)) * k_m) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.8e-112], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-112

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.5%

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

   if 1.8e-112 < k

   1. Initial program 24.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 96.3%

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

   11. Applied rewrites 96.3%

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

Alternative 4: 94.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\cos k\_m \cdot 2}{k\_m} \cdot \frac{\ell}{{\sin k\_m}^{2} \cdot t}\right) \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e-9)
   (/
    2.0
    (*
     (*
      k_m
      (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
     (/ k_m (* l (cos k_m)))))
   (/
    (* (* (/ (* (cos k_m) 2.0) k_m) (/ l (* (pow (sin k_m) 2.0) t))) l)
    k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-9) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * (k_m / (l * cos(k_m))));
	} else {
		tmp = ((((cos(k_m) * 2.0) / k_m) * (l / (pow(sin(k_m), 2.0) * t))) * l) / k_m;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-9)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * Float64(k_m / Float64(l * cos(k_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) * 2.0) / k_m) * Float64(l / Float64((sin(k_m) ^ 2.0) * t))) * l) / k_m);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e-9], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000012e-9

   1. Initial program 37.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 83.1%

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

   if 2.00000000000000012e-9 < k

   1. Initial program 15.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 91.6%

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

Alternative 5: 95.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 90.6

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

5. Applied rewrites 90.6%

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

7. Applied rewrites 94.7%

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

9. Applied rewrites 94.1%

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

11. Applied rewrites 96.8%

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

Alternative 6: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k\_m \cdot 2\right) \cdot \ell}{k\_m} \cdot \frac{\ell}{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.9e-10)
   (/
    2.0
    (*
     (*
      k_m
      (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
     (/ k_m (* l (cos k_m)))))
   (*
    (/ (* (* (cos k_m) 2.0) l) k_m)
    (/ l (* (* (pow (sin k_m) 2.0) t) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.9e-10) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * (k_m / (l * cos(k_m))));
	} else {
		tmp = (((cos(k_m) * 2.0) * l) / k_m) * (l / ((pow(sin(k_m), 2.0) * t) * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.9e-10)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * Float64(k_m / Float64(l * cos(k_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * 2.0) * l) / k_m) * Float64(l / Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.9e-10], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.89999999999999981e-10

   1. Initial program 37.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 83.1%

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

   if 2.89999999999999981e-10 < k

   1. Initial program 15.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 91.5%

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

Alternative 7: 89.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot 2\\ t_2 := {\sin k\_m}^{2} \cdot t\\ \mathbf{if}\;k\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \frac{t\_1 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{\ell \cdot \ell}{k\_m}}{t\_2 \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) 2.0)) (t_2 (* (pow (sin k_m) 2.0) t)))
   (if (<= k_m 1.32e+154)
     (* (/ l t_2) (/ (* t_1 l) (* k_m k_m)))
     (* t_1 (/ (/ (* l l) k_m) (* t_2 k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * 2.0;
	double t_2 = pow(sin(k_m), 2.0) * t;
	double tmp;
	if (k_m <= 1.32e+154) {
		tmp = (l / t_2) * ((t_1 * l) / (k_m * k_m));
	} else {
		tmp = t_1 * (((l * l) / k_m) / (t_2 * 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) :: t_2
    real(8) :: tmp
    t_1 = cos(k_m) * 2.0d0
    t_2 = (sin(k_m) ** 2.0d0) * t
    if (k_m <= 1.32d+154) then
        tmp = (l / t_2) * ((t_1 * l) / (k_m * k_m))
    else
        tmp = t_1 * (((l * l) / k_m) / (t_2 * 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.cos(k_m) * 2.0;
	double t_2 = Math.pow(Math.sin(k_m), 2.0) * t;
	double tmp;
	if (k_m <= 1.32e+154) {
		tmp = (l / t_2) * ((t_1 * l) / (k_m * k_m));
	} else {
		tmp = t_1 * (((l * l) / k_m) / (t_2 * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * 2.0
	t_2 = math.pow(math.sin(k_m), 2.0) * t
	tmp = 0
	if k_m <= 1.32e+154:
		tmp = (l / t_2) * ((t_1 * l) / (k_m * k_m))
	else:
		tmp = t_1 * (((l * l) / k_m) / (t_2 * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * 2.0)
	t_2 = Float64((sin(k_m) ^ 2.0) * t)
	tmp = 0.0
	if (k_m <= 1.32e+154)
		tmp = Float64(Float64(l / t_2) * Float64(Float64(t_1 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(l * l) / k_m) / Float64(t_2 * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * 2.0;
	t_2 = (sin(k_m) ^ 2.0) * t;
	tmp = 0.0;
	if (k_m <= 1.32e+154)
		tmp = (l / t_2) * ((t_1 * l) / (k_m * k_m));
	else
		tmp = t_1 * (((l * l) / k_m) / (t_2 * 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[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k$95$m, 1.32e+154], N[(N[(l / t$95$2), $MachinePrecision] * N[(N[(t$95$1 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t$95$2 * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.31999999999999998e154

   1. Initial program 34.3%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 78.9%

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 90.7%

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

   if 1.31999999999999998e154 < k

   1. Initial program 22.9%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 68.7%

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

Alternative 8: 93.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 90.6

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

5. Applied rewrites 90.6%

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

7. Applied rewrites 94.7%

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

Alternative 9: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{t \cdot {\left(\sin k\_m \cdot k\_m\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.4e-8)
   (/
    2.0
    (*
     (*
      k_m
      (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
     (/ k_m (* l (cos k_m)))))
   (* l (* (* (cos k_m) 2.0) (/ l (* t (pow (* (sin k_m) k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e-8) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * (k_m / (l * cos(k_m))));
	} else {
		tmp = l * ((cos(k_m) * 2.0) * (l / (t * pow((sin(k_m) * k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.4e-8)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * Float64(k_m / Float64(l * cos(k_m)))));
	else
		tmp = Float64(l * Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(t * (Float64(sin(k_m) * k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.4e-8], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(t * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-8

   1. Initial program 37.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 83.1%

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

   if 1.4e-8 < k

   1. Initial program 15.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 68.7%

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

Alternative 10: 86.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot 2\right) \cdot \ell\right) \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.75e-8)
   (/
    2.0
    (*
     (*
      k_m
      (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
     (/ k_m (* l (cos k_m)))))
   (/
    (* (* (* (cos k_m) 2.0) l) l)
    (* (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.75e-8) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * (k_m / (l * cos(k_m))));
	} else {
		tmp = (((cos(k_m) * 2.0) * l) * l) / ((((0.5 - (0.5 * cos((k_m + k_m)))) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.75e-8)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * Float64(k_m / Float64(l * cos(k_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * 2.0) * l) * l) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t) * 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, 1.75e-8], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75000000000000012e-8

   1. Initial program 37.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 83.1%

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

   if 1.75000000000000012e-8 < k

   1. Initial program 15.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 65.0%

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

Alternative 11: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell \cdot \cos k\_m}\\ \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \frac{\left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m}{\ell}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \left(\frac{t}{\ell} \cdot \left(k\_m \cdot k\_m\right)\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* l (cos k_m)))))
   (if (<= k_m 1.8e-112)
     (/
      2.0
      (*
       (*
        k_m
        (/ (* (* (fma (* -0.3333333333333333 t) (* k_m k_m) t) k_m) k_m) l))
       t_1))
     (/ 2.0 (* (* k_m (* (/ t l) (* k_m k_m))) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (l * cos(k_m));
	double tmp;
	if (k_m <= 1.8e-112) {
		tmp = 2.0 / ((k_m * (((fma((-0.3333333333333333 * t), (k_m * k_m), t) * k_m) * k_m) / l)) * t_1);
	} else {
		tmp = 2.0 / ((k_m * ((t / l) * (k_m * k_m))) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(l * cos(k_m)))
	tmp = 0.0
	if (k_m <= 1.8e-112)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(Float64(fma(Float64(-0.3333333333333333 * t), Float64(k_m * k_m), t) * k_m) * k_m) / l)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(t / l) * Float64(k_m * k_m))) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.8e-112], N[(2.0 / N[(N[(k$95$m * N[(N[(N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(N[(t / l), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-112

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.5%

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

   if 1.8e-112 < k

   1. Initial program 24.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 63.4%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 70.1%

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

Alternative 12: 75.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}}{k\_m}}{k\_m} \cdot \left(\ell + \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \left(\frac{t}{\ell} \cdot \left(k\_m \cdot k\_m\right)\right)\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.8e-112)
   (* (/ (/ (/ l (* (* k_m k_m) t)) k_m) k_m) (+ l l))
   (/ 2.0 (* (* k_m (* (/ t l) (* k_m k_m))) (/ k_m (* l (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-112) {
		tmp = (((l / ((k_m * k_m) * t)) / k_m) / k_m) * (l + l);
	} else {
		tmp = 2.0 / ((k_m * ((t / l) * (k_m * k_m))) * (k_m / (l * cos(k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d-112) then
        tmp = (((l / ((k_m * k_m) * t)) / k_m) / k_m) * (l + l)
    else
        tmp = 2.0d0 / ((k_m * ((t / l) * (k_m * k_m))) * (k_m / (l * cos(k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-112) {
		tmp = (((l / ((k_m * k_m) * t)) / k_m) / k_m) * (l + l);
	} else {
		tmp = 2.0 / ((k_m * ((t / l) * (k_m * k_m))) * (k_m / (l * Math.cos(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.8e-112:
		tmp = (((l / ((k_m * k_m) * t)) / k_m) / k_m) * (l + l)
	else:
		tmp = 2.0 / ((k_m * ((t / l) * (k_m * k_m))) * (k_m / (l * math.cos(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-112)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(k_m * k_m) * t)) / k_m) / k_m) * Float64(l + l));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(t / l) * Float64(k_m * k_m))) * Float64(k_m / Float64(l * cos(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e-112)
		tmp = (((l / ((k_m * k_m) * t)) / k_m) / k_m) * (l + l);
	else
		tmp = 2.0 / ((k_m * ((t / l) * (k_m * k_m))) * (k_m / (l * cos(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.8e-112], N[(N[(N[(N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(N[(t / l), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-112

   1. Initial program 37.2%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 77.1%

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

   11. Applied rewrites 80.1%

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

   if 1.8e-112 < k

   1. Initial program 24.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 63.4%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 70.1%

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

Alternative 13: 74.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.28:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{k\_m}}{t}}{k\_m \cdot k\_m}}{k\_m} \cdot \left(\ell + \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m} \cdot \frac{\ell \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.28)
   (* (/ (/ (/ (/ l k_m) t) (* k_m k_m)) k_m) (+ l l))
   (* (/ (* 2.0 (cos k_m)) (* (* (* k_m k_m) t) k_m)) (/ (* l l) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.28) {
		tmp = ((((l / k_m) / t) / (k_m * k_m)) / k_m) * (l + l);
	} else {
		tmp = ((2.0 * cos(k_m)) / (((k_m * k_m) * t) * k_m)) * ((l * l) / 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 <= 0.28d0) then
        tmp = ((((l / k_m) / t) / (k_m * k_m)) / k_m) * (l + l)
    else
        tmp = ((2.0d0 * cos(k_m)) / (((k_m * k_m) * t) * k_m)) * ((l * l) / 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 <= 0.28) {
		tmp = ((((l / k_m) / t) / (k_m * k_m)) / k_m) * (l + l);
	} else {
		tmp = ((2.0 * Math.cos(k_m)) / (((k_m * k_m) * t) * k_m)) * ((l * l) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.28:
		tmp = ((((l / k_m) / t) / (k_m * k_m)) / k_m) * (l + l)
	else:
		tmp = ((2.0 * math.cos(k_m)) / (((k_m * k_m) * t) * k_m)) * ((l * l) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.28)
		tmp = Float64(Float64(Float64(Float64(Float64(l / k_m) / t) / Float64(k_m * k_m)) / k_m) * Float64(l + l));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(Float64(k_m * k_m) * t) * k_m)) * Float64(Float64(l * l) / 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 <= 0.28)
		tmp = ((((l / k_m) / t) / (k_m * k_m)) / k_m) * (l + l);
	else
		tmp = ((2.0 * cos(k_m)) / (((k_m * k_m) * t) * k_m)) * ((l * l) / 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, 0.28], N[(N[(N[(N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.28000000000000003

   1. Initial program 37.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 78.1%

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

   11. Applied rewrites 82.6%

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

   if 0.28000000000000003 < k

   1. Initial program 15.3%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 52.6%

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

Alternative 14: 72.9% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 75.7%

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

Alternative 15: 73.5% accurate, 9.6× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * 2.0) / ((k_m * k_m) * t)) * (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 = ((l * 2.0d0) / ((k_m * k_m) * t)) * (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 ((l * 2.0) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 75.2%

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

Alternative 16: 73.0% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 72.0%

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

11. Applied rewrites 74.9%

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

Alternative 17: 70.8% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 72.0%

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

11. Applied rewrites 72.0%

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

Alternative 18: 70.8% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 72.0%

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

Reproduce

herbie shell --seed 2024343 
(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))))