Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 97.8%

Time: 13.0s

Alternatives: 17

Speedup: 11.0×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{k\_m}{\cos k\_m}}{\ell}\\ \mathbf{if}\;k\_m \leq 4 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(k\_m \cdot k\_m, -0.16666666666666666, 1\right) \cdot \left(\frac{t \cdot k\_m}{\ell} \cdot k\_m\right)\right) \cdot \sin k\_m\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot {\sin k\_m}^{2}\right) \cdot \left(\frac{k\_m}{\ell} \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ k_m (cos k_m)) l)))
   (if (<= k_m 4e-133)
     (/
      2.0
      (*
       (*
        (* (fma (* k_m k_m) -0.16666666666666666 1.0) (* (/ (* t k_m) l) k_m))
        (sin k_m))
       t_1))
     (/ 2.0 (* (* t_1 (pow (sin k_m) 2.0)) (* (/ k_m l) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / cos(k_m)) / l;
	double tmp;
	if (k_m <= 4e-133) {
		tmp = 2.0 / (((fma((k_m * k_m), -0.16666666666666666, 1.0) * (((t * k_m) / l) * k_m)) * sin(k_m)) * t_1);
	} else {
		tmp = 2.0 / ((t_1 * pow(sin(k_m), 2.0)) * ((k_m / l) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / cos(k_m)) / l)
	tmp = 0.0
	if (k_m <= 4e-133)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k_m * k_m), -0.16666666666666666, 1.0) * Float64(Float64(Float64(t * k_m) / l) * k_m)) * sin(k_m)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * (sin(k_m) ^ 2.0)) * Float64(Float64(k_m / l) * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 4e-133], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(t * k$95$m), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.0000000000000003e-133

   1. Initial program 41.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 95.7%

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 76.3%

       \[\leadsto \frac{2}{\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \left(\frac{t \cdot k}{\ell} \cdot k\right)\right) \cdot \sin k\right) \cdot \frac{\frac{\color{blue}{k}}{\cos k}}{\ell}} \]

   if 4.0000000000000003e-133 < k

   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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 86.8%

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

   7. Applied rewrites 97.5%

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

   9. Applied rewrites 98.5%

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

Alternative 2: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{k\_m}{\cos k\_m}}{\ell}\\ \mathbf{if}\;k\_m \leq 0.0003:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(k\_m \cdot k\_m, -0.16666666666666666, 1\right) \cdot \left(\frac{t \cdot k\_m}{\ell} \cdot k\_m\right)\right) \cdot \sin k\_m\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot \left({\sin k\_m}^{2} \cdot t\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ k_m (cos k_m)) l)))
   (if (<= k_m 0.0003)
     (/
      2.0
      (*
       (*
        (* (fma (* k_m k_m) -0.16666666666666666 1.0) (* (/ (* t k_m) l) k_m))
        (sin k_m))
       t_1))
     (/ 2.0 (* (* t_1 (* (pow (sin k_m) 2.0) t)) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / cos(k_m)) / l;
	double tmp;
	if (k_m <= 0.0003) {
		tmp = 2.0 / (((fma((k_m * k_m), -0.16666666666666666, 1.0) * (((t * k_m) / l) * k_m)) * sin(k_m)) * t_1);
	} else {
		tmp = 2.0 / ((t_1 * (pow(sin(k_m), 2.0) * t)) * (k_m / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / cos(k_m)) / l)
	tmp = 0.0
	if (k_m <= 0.0003)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k_m * k_m), -0.16666666666666666, 1.0) * Float64(Float64(Float64(t * k_m) / l) * k_m)) * sin(k_m)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64((sin(k_m) ^ 2.0) * t)) * Float64(k_m / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0003], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(t * k$95$m), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999974e-4

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 95.8%

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 79.6%

       \[\leadsto \frac{2}{\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \left(\frac{t \cdot k}{\ell} \cdot k\right)\right) \cdot \sin k\right) \cdot \frac{\frac{\color{blue}{k}}{\cos k}}{\ell}} \]

   if 2.99999999999999974e-4 < k

   1. Initial program 35.1%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 97.9%

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

Alternative 3: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 89.3%

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

7. Applied rewrites 96.4%

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

9. Applied rewrites 98.4%

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

Alternative 4: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{k\_m}{\cos k\_m}}{\ell}\\ \mathbf{if}\;k\_m \leq 0.00365:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(k\_m \cdot k\_m, -0.16666666666666666, 1\right) \cdot \left(\frac{t \cdot k\_m}{\ell} \cdot k\_m\right)\right) \cdot \sin k\_m\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ k_m (cos k_m)) l)))
   (if (<= k_m 0.00365)
     (/
      2.0
      (*
       (*
        (* (fma (* k_m k_m) -0.16666666666666666 1.0) (* (/ (* t k_m) l) k_m))
        (sin k_m))
       t_1))
     (/ 2.0 (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) (* t (/ k_m l))) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / cos(k_m)) / l;
	double tmp;
	if (k_m <= 0.00365) {
		tmp = 2.0 / (((fma((k_m * k_m), -0.16666666666666666, 1.0) * (((t * k_m) / l) * k_m)) * sin(k_m)) * t_1);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((k_m + k_m)))) * (t * (k_m / l))) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / cos(k_m)) / l)
	tmp = 0.0
	if (k_m <= 0.00365)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(k_m * k_m), -0.16666666666666666, 1.0) * Float64(Float64(Float64(t * k_m) / l) * k_m)) * sin(k_m)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * Float64(t * Float64(k_m / l))) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 0.00365], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(N[(t * k$95$m), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00365000000000000003

   1. Initial program 39.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 95.8%

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 79.6%

       \[\leadsto \frac{2}{\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \left(\frac{t \cdot k}{\ell} \cdot k\right)\right) \cdot \sin k\right) \cdot \frac{\frac{\color{blue}{k}}{\cos k}}{\ell}} \]

   if 0.00365000000000000003 < k

   1. Initial program 35.1%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 98.0%

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

   9. Applied rewrites 97.7%

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

Alternative 5: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-9)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (/
    2.0
    (*
     (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) (* t (/ k_m l)))
     (/ (/ k_m (cos k_m)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((k_m + k_m)))) * (t * (k_m / l))) * ((k_m / cos(k_m)) / l));
	}
	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 <= 5d-9) then
        tmp = ((l / (k_m * k_m)) / t) * ((2.0d0 * l) / (k_m * k_m))
    else
        tmp = 2.0d0 / (((0.5d0 - (0.5d0 * cos((k_m + k_m)))) * (t * (k_m / l))) * ((k_m / cos(k_m)) / l))
    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 <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * Math.cos((k_m + k_m)))) * (t * (k_m / l))) * ((k_m / Math.cos(k_m)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5e-9:
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m))
	else:
		tmp = 2.0 / (((0.5 - (0.5 * math.cos((k_m + k_m)))) * (t * (k_m / l))) * ((k_m / math.cos(k_m)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-9)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * Float64(t * Float64(k_m / l))) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e-9)
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	else
		tmp = 2.0 / (((0.5 - (0.5 * cos((k_m + k_m)))) * (t * (k_m / l))) * ((k_m / cos(k_m)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-9], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000001e-9

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 77.0%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 97.4%

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

Alternative 6: 94.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\cos \left(2 \cdot k\_m\right), -0.5, 0.5\right) \cdot \frac{t}{\ell}\right) \cdot k\_m\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-9)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (/
    2.0
    (*
     (* (* (fma (cos (* 2.0 k_m)) -0.5 0.5) (/ t l)) k_m)
     (/ (/ k_m (cos k_m)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = 2.0 / (((fma(cos((2.0 * k_m)), -0.5, 0.5) * (t / l)) * k_m) * ((k_m / cos(k_m)) / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-9)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(cos(Float64(2.0 * k_m)), -0.5, 0.5) * Float64(t / l)) * k_m) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-9], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000001e-9

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 77.0%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 93.6%

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

Alternative 7: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t}}{k\_m}}{k\_m} \cdot \ell\right) \cdot \left(\cos k\_m \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-9)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (*
    (* (/ (/ (/ 2.0 (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t)) k_m) k_m) l)
    (* (cos k_m) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((((2.0 / ((0.5 - (0.5 * cos((k_m + k_m)))) * t)) / k_m) / k_m) * l) * (cos(k_m) * l);
	}
	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 <= 5d-9) then
        tmp = ((l / (k_m * k_m)) / t) * ((2.0d0 * l) / (k_m * k_m))
    else
        tmp = ((((2.0d0 / ((0.5d0 - (0.5d0 * cos((k_m + k_m)))) * t)) / k_m) / k_m) * l) * (cos(k_m) * l)
    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 <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((((2.0 / ((0.5 - (0.5 * Math.cos((k_m + k_m)))) * t)) / k_m) / k_m) * l) * (Math.cos(k_m) * l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5e-9:
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m))
	else:
		tmp = ((((2.0 / ((0.5 - (0.5 * math.cos((k_m + k_m)))) * t)) / k_m) / k_m) * l) * (math.cos(k_m) * l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-9)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t)) / k_m) / k_m) * l) * Float64(cos(k_m) * l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e-9)
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	else
		tmp = ((((2.0 / ((0.5 - (0.5 * cos((k_m + k_m)))) * t)) / k_m) / k_m) * l) * (cos(k_m) * l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-9], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000001e-9

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 77.0%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 82.4%

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

   9. Applied rewrites 81.9%

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

Alternative 8: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t}}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-9)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (*
    (* (* (cos k_m) l) l)
    (/ (/ (/ 2.0 (* (- 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 <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((cos(k_m) * l) * l) * (((2.0 / ((0.5 - (0.5 * cos((k_m + k_m)))) * t)) / k_m) / k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d-9) then
        tmp = ((l / (k_m * k_m)) / t) * ((2.0d0 * l) / (k_m * k_m))
    else
        tmp = ((cos(k_m) * l) * l) * (((2.0d0 / ((0.5d0 - (0.5d0 * cos((k_m + k_m)))) * t)) / k_m) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((Math.cos(k_m) * l) * l) * (((2.0 / ((0.5 - (0.5 * Math.cos((k_m + k_m)))) * t)) / k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5e-9:
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m))
	else:
		tmp = ((math.cos(k_m) * l) * l) * (((2.0 / ((0.5 - (0.5 * math.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 <= 5e-9)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) * l) * Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t)) / k_m) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e-9)
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	else
		tmp = ((cos(k_m) * l) * l) * (((2.0 / ((0.5 - (0.5 * cos((k_m + k_m)))) * t)) / k_m) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-9], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000001e-9

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 77.0%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 73.8%

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

Alternative 9: 82.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \frac{\mathsf{fma}\left(\cos \left(2 \cdot k\_m\right), -0.5, 0.5\right)}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-9)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (/
    2.0
    (*
     (* (* k_m k_m) t)
     (/ (fma (cos (* 2.0 k_m)) -0.5 0.5) (* (* (cos k_m) l) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-9) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = 2.0 / (((k_m * k_m) * t) * (fma(cos((2.0 * k_m)), -0.5, 0.5) / ((cos(k_m) * l) * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-9)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t) * Float64(fma(cos(Float64(2.0 * k_m)), -0.5, 0.5) / Float64(Float64(cos(k_m) * l) * l))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-9], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000001e-9

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 77.0%

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

   if 5.0000000000000001e-9 < k

   1. Initial program 33.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 65.9%

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

Alternative 10: 77.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\frac{0.6666666666666666}{t}}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.6)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (* (* (* (cos k_m) l) l) (/ (/ (/ 0.6666666666666666 t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((cos(k_m) * l) * l) * (((0.6666666666666666 / t) / k_m) / k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.6d0) then
        tmp = ((l / (k_m * k_m)) / t) * ((2.0d0 * l) / (k_m * k_m))
    else
        tmp = ((cos(k_m) * l) * l) * (((0.6666666666666666d0 / t) / k_m) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((Math.cos(k_m) * l) * l) * (((0.6666666666666666 / t) / k_m) / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.6:
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m))
	else:
		tmp = ((math.cos(k_m) * l) * l) * (((0.6666666666666666 / t) / k_m) / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) * l) * Float64(Float64(Float64(0.6666666666666666 / t) / k_m) / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.6)
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	else
		tmp = ((cos(k_m) * l) * l) * (((0.6666666666666666 / t) / k_m) / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.6], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5999999999999996

   1. Initial program 39.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 71.5%

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

   9. Applied rewrites 74.3%

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

   11. Applied rewrites 77.2%

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

   if 6.5999999999999996 < k

   1. Initial program 35.1%

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

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 55.3%

       \[\leadsto \left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\frac{0.6666666666666666}{t}}{k}}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m \cdot k\_m}}{t} \cdot \frac{2 \cdot \ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.5, k\_m \cdot k\_m, 1\right) \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\frac{\frac{\mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, 0.6666666666666666, \frac{2}{t}\right)}{k\_m}}{k\_m}}{k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 6e+207)
   (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* k_m k_m)))
   (*
    (* (* (fma -0.5 (* k_m k_m) 1.0) l) l)
    (/
     (/
      (/ (/ (fma (/ (* k_m k_m) t) 0.6666666666666666 (/ 2.0 t)) k_m) k_m)
      k_m)
     k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 6e+207) {
		tmp = ((l / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
	} else {
		tmp = ((fma(-0.5, (k_m * k_m), 1.0) * l) * l) * ((((fma(((k_m * k_m) / t), 0.6666666666666666, (2.0 / t)) / k_m) / k_m) / k_m) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 6e+207)
		tmp = Float64(Float64(Float64(l / Float64(k_m * k_m)) / t) * Float64(Float64(2.0 * l) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(fma(-0.5, Float64(k_m * k_m), 1.0) * l) * l) * Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) / t), 0.6666666666666666, Float64(2.0 / t)) / k_m) / k_m) / k_m) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 6e+207], N[(N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * 0.6666666666666666 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.99999999999999966e207

   1. Initial program 38.3%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 68.1%

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

   11. Applied rewrites 70.2%

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

   if 5.99999999999999966e207 < l

   1. Initial program 36.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 52.8%

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

   9. Taylor expanded in k around 0

      \[\leadsto \left(\left(\ell + \frac{-1}{2} \cdot \left({k}^{2} \cdot \ell\right)\right) \cdot \ell\right) \cdot \frac{\frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, \frac{2}{3}, \frac{2}{t}\right)}{k}}{k}}}{k}}{k} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.3%

       \[\leadsto \left(\left(\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, 0.6666666666666666, \frac{2}{t}\right)}{k}}{k}}}{k}}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.9% accurate, 8.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (/ l (* k_m k_m)) t) (/ (* 2.0 l) (* 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) * ((2.0 * 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 / (k_m * k_m)) / t) * ((2.0d0 * 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 / (k_m * k_m)) / t) * ((2.0 * l) / (k_m * k_m));
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 64.3%

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

9. Applied rewrites 66.0%

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

11. Applied rewrites 68.0%

    \[\leadsto \frac{\frac{\ell}{k \cdot k}}{t} \cdot \color{blue}{\frac{2 \cdot \ell}{k \cdot k}} \]
12. Add Preprocessing

Alternative 13: 73.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((2.0d0 * l) / k_m) * ((l / (k_m * k_m)) / (t * k_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 64.3%

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

9. Applied rewrites 66.0%

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

11. Applied rewrites 67.3%

    \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{t \cdot k}} \]
12. Add Preprocessing

Alternative 14: 71.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * ((l / (k_m * k_m)) / ((t * k_m) * k_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 64.3%

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

9. Applied rewrites 66.0%

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

11. Applied rewrites 66.0%

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

Alternative 15: 71.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * ((l / (k_m * k_m)) / ((k_m * k_m) * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 64.3%

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

9. Applied rewrites 66.0%

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

Alternative 16: 70.9% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 64.3%

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

9. Applied rewrites 66.0%

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

11. Applied rewrites 65.6%

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

Alternative 17: 69.6% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * (l / (((k_m * k_m) * t) * (k_m * k_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 64.3%

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

9. Applied rewrites 66.0%

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

11. Applied rewrites 63.4%

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

12. Final simplification 63.4%

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

Reproduce

herbie shell --seed 2024318 
(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))))