Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 98.4%

Time: 13.6s

Alternatives: 16

Speedup: 9.6×

• 38.5% 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: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-118}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \frac{k\_m}{\ell}\right) \cdot \left(\frac{k\_m}{\ell} \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1e-118)
   (/ 2.0 (* (* (pow (/ k_m l) 2.0) k_m) (* k_m t)))
   (/ 2.0 (* (* (* (tan k_m) (sin k_m)) (/ k_m l)) (* (/ k_m l) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-118) {
		tmp = 2.0 / ((pow((k_m / l), 2.0) * k_m) * (k_m * t));
	} else {
		tmp = 2.0 / (((tan(k_m) * sin(k_m)) * (k_m / l)) * ((k_m / l) * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1d-118) then
        tmp = 2.0d0 / ((((k_m / l) ** 2.0d0) * k_m) * (k_m * t))
    else
        tmp = 2.0d0 / (((tan(k_m) * sin(k_m)) * (k_m / l)) * ((k_m / l) * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-118) {
		tmp = 2.0 / ((Math.pow((k_m / l), 2.0) * k_m) * (k_m * t));
	} else {
		tmp = 2.0 / (((Math.tan(k_m) * Math.sin(k_m)) * (k_m / l)) * ((k_m / l) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1e-118:
		tmp = 2.0 / ((math.pow((k_m / l), 2.0) * k_m) * (k_m * t))
	else:
		tmp = 2.0 / (((math.tan(k_m) * math.sin(k_m)) * (k_m / l)) * ((k_m / l) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1e-118)
		tmp = Float64(2.0 / Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * Float64(k_m * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(k_m / l)) * Float64(Float64(k_m / l) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1e-118)
		tmp = 2.0 / ((((k_m / l) ^ 2.0) * k_m) * (k_m * t));
	else
		tmp = 2.0 / (((tan(k_m) * sin(k_m)) * (k_m / l)) * ((k_m / l) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1e-118], N[(2.0 / N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999985e-119

   1. Initial program 35.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 64.1%

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

   9. Applied rewrites 74.8%

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

   if 9.99999999999999985e-119 < k

   1. Initial program 33.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 90.0%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 91.0%

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

   11. Applied rewrites 99.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-118}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot k\right) \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \sin k\right) \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \left(k\_m \cdot t\right)\right) \cdot k\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.2e-79)
   (/ 2.0 (* (* (pow (/ k_m l) 2.0) k_m) (* k_m t)))
   (/ 2.0 (/ (* (* (* (tan k_m) (sin k_m)) (* k_m t)) k_m) (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-79) {
		tmp = 2.0 / ((pow((k_m / l), 2.0) * k_m) * (k_m * t));
	} else {
		tmp = 2.0 / ((((tan(k_m) * sin(k_m)) * (k_m * t)) * k_m) / (l * 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 <= 1.2d-79) then
        tmp = 2.0d0 / ((((k_m / l) ** 2.0d0) * k_m) * (k_m * t))
    else
        tmp = 2.0d0 / ((((tan(k_m) * sin(k_m)) * (k_m * t)) * k_m) / (l * 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 <= 1.2e-79) {
		tmp = 2.0 / ((Math.pow((k_m / l), 2.0) * k_m) * (k_m * t));
	} else {
		tmp = 2.0 / ((((Math.tan(k_m) * Math.sin(k_m)) * (k_m * t)) * k_m) / (l * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.2e-79:
		tmp = 2.0 / ((math.pow((k_m / l), 2.0) * k_m) * (k_m * t))
	else:
		tmp = 2.0 / ((((math.tan(k_m) * math.sin(k_m)) * (k_m * t)) * k_m) / (l * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.2e-79)
		tmp = Float64(2.0 / Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * Float64(k_m * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(k_m * t)) * k_m) / Float64(l * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.2e-79)
		tmp = 2.0 / ((((k_m / l) ^ 2.0) * k_m) * (k_m * t));
	else
		tmp = 2.0 / ((((tan(k_m) * sin(k_m)) * (k_m * t)) * k_m) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.2e-79], N[(2.0 / N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000003e-79

   1. Initial program 36.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 66.8

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 64.1%

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

   9. Applied rewrites 75.3%

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

   if 1.20000000000000003e-79 < k

   1. Initial program 32.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 90.8%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 90.7%

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

   11. Applied rewrites 70.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot k\right) \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot t\right)\right) \cdot k}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 91.3%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 91.9%

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

11. Applied rewrites 98.6%

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

12. Final simplification 98.6%

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

Alternative 4: 73.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.2e-79)
   (/ 2.0 (* (* (pow (/ k_m l) 2.0) k_m) (* k_m t)))
   (/ 2.0 (/ (* (* (* (* k_m k_m) t) k_m) k_m) (* (* (cos k_m) l) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-79) {
		tmp = 2.0 / ((pow((k_m / l), 2.0) * k_m) * (k_m * t));
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) * k_m) * k_m) / ((cos(k_m) * l) * 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 <= 1.2d-79) then
        tmp = 2.0d0 / ((((k_m / l) ** 2.0d0) * k_m) * (k_m * t))
    else
        tmp = 2.0d0 / (((((k_m * k_m) * t) * k_m) * k_m) / ((cos(k_m) * l) * 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 <= 1.2e-79) {
		tmp = 2.0 / ((Math.pow((k_m / l), 2.0) * k_m) * (k_m * t));
	} else {
		tmp = 2.0 / (((((k_m * k_m) * t) * k_m) * k_m) / ((Math.cos(k_m) * l) * l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.2e-79:
		tmp = 2.0 / ((math.pow((k_m / l), 2.0) * k_m) * (k_m * t))
	else:
		tmp = 2.0 / (((((k_m * k_m) * t) * k_m) * k_m) / ((math.cos(k_m) * l) * l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.2e-79)
		tmp = Float64(2.0 / Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * Float64(k_m * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * k_m) * k_m) / Float64(Float64(cos(k_m) * l) * l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.2e-79)
		tmp = 2.0 / ((((k_m / l) ^ 2.0) * k_m) * (k_m * t));
	else
		tmp = 2.0 / (((((k_m * k_m) * t) * k_m) * k_m) / ((cos(k_m) * l) * l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.2e-79], N[(2.0 / N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000003e-79

   1. Initial program 36.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 66.8

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 64.1%

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

   9. Applied rewrites 75.3%

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

   if 1.20000000000000003e-79 < k

   1. Initial program 32.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 90.8%

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 56.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot k\right) \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot k\right) \cdot k}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 62.7

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

5. Applied rewrites 62.7%

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

7. Applied rewrites 60.9%

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

9. Applied rewrites 68.9%

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

10. Final simplification 68.9%

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

Alternative 6: 74.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5400:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m \cdot t}{\ell} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k\_m \cdot k\_m, 0.16666666666666666\right), k\_m \cdot k\_m, 1\right) \cdot k\_m\right) \cdot k\_m\right)\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k\_m} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5400.0)
   (/
    2.0
    (*
     (*
      (/ (* k_m t) l)
      (*
       (*
        (fma
         (fma 0.08611111111111111 (* k_m k_m) 0.16666666666666666)
         (* k_m k_m)
         1.0)
        k_m)
       k_m))
     (/ k_m l)))
   (* (/ -0.3333333333333333 k_m) (/ (/ (* l l) t) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5400.0) {
		tmp = 2.0 / ((((k_m * t) / l) * ((fma(fma(0.08611111111111111, (k_m * k_m), 0.16666666666666666), (k_m * k_m), 1.0) * k_m) * k_m)) * (k_m / l));
	} else {
		tmp = (-0.3333333333333333 / k_m) * (((l * l) / t) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5400.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * t) / l) * Float64(Float64(fma(fma(0.08611111111111111, Float64(k_m * k_m), 0.16666666666666666), Float64(k_m * k_m), 1.0) * k_m) * k_m)) * Float64(k_m / l)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / k_m) * Float64(Float64(Float64(l * l) / t) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5400.0], N[(2.0 / N[(N[(N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.08611111111111111 * N[(k$95$m * k$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / k$95$m), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5400

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 95.5%

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

   9. Applied rewrites 92.7%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{k}{\ell} \cdot \left(\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right) \cdot \frac{\color{blue}{k \cdot t}}{\ell}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 73.8%

       \[\leadsto \frac{2}{\frac{k}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot k\right) \cdot k\right) \cdot \frac{\color{blue}{k \cdot t}}{\ell}\right)} \]

   if 5400 < k

   1. Initial program 32.6%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. div-add N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}}}{{k}^{4}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 \cdot \ell, \ell, \left(-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(k \cdot k\right)\right)}{{k}^{4} \cdot t}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.3%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5400:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5400:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m \cdot t}{\ell}\right) \cdot \frac{k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k\_m} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5400.0)
   (/
    2.0
    (*
     (*
      (* (* (fma 0.16666666666666666 (* k_m k_m) 1.0) k_m) k_m)
      (/ (* k_m t) l))
     (/ k_m l)))
   (* (/ -0.3333333333333333 k_m) (/ (/ (* l l) t) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5400.0) {
		tmp = 2.0 / ((((fma(0.16666666666666666, (k_m * k_m), 1.0) * k_m) * k_m) * ((k_m * t) / l)) * (k_m / l));
	} else {
		tmp = (-0.3333333333333333 / k_m) * (((l * l) / t) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5400.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(0.16666666666666666, Float64(k_m * k_m), 1.0) * k_m) * k_m) * Float64(Float64(k_m * t) / l)) * Float64(k_m / l)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / k_m) * Float64(Float64(Float64(l * l) / t) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5400.0], N[(2.0 / N[(N[(N[(N[(N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / k$95$m), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5400

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 91.9%

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

   7. Applied rewrites 95.5%

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

   9. Applied rewrites 92.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 73.7%

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

   if 5400 < k

   1. Initial program 32.6%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. div-add N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}}}{{k}^{4}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. div-add-rev N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 \cdot \ell, \ell, \left(-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(k \cdot k\right)\right)}{{k}^{4} \cdot t}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.3%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5400:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot k\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-277}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot 2}{k\_m \cdot k\_m}}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= (* l l) 1e-277)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ (/ (* (/ (* l l) t) 2.0) (* k_m k_m)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if ((l * l) <= 1e-277) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k_m / l) * k_m
    if ((l * l) <= 1d-277) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = ((((l * l) / t) * 2.0d0) / (k_m * k_m)) / (k_m * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if ((l * l) <= 1e-277) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	tmp = 0
	if (l * l) <= 1e-277:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (Float64(l * l) <= 1e-277)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t) * 2.0) / Float64(k_m * k_m)) / Float64(k_m * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 0.0;
	if ((l * l) <= 1e-277)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-277], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999969e-278

   1. Initial program 27.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 66.2%

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

   9. Applied rewrites 87.2%

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

   if 9.99999999999999969e-278 < (*.f64 l l)

   1. Initial program 37.7%

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

   3. 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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 24.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 58.8%

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

   13. Applied rewrites 61.4%

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

Alternative 9: 64.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m \cdot k\_m}{\ell \cdot \ell} \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot 2}{k\_m \cdot k\_m}}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1.1e-132)
   (/ 2.0 (* (* (/ (* k_m k_m) (* l l)) (* k_m k_m)) t))
   (/ (/ (* (/ (* l l) t) 2.0) (* 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 <= 1.1e-132) {
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t);
	} else {
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 1.1d-132) then
        tmp = 2.0d0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t)
    else
        tmp = ((((l * l) / t) * 2.0d0) / (k_m * k_m)) / (k_m * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.1e-132) {
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t);
	} else {
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1.1e-132:
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t)
	else:
		tmp = ((((l * l) / t) * 2.0) / (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 <= 1.1e-132)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / Float64(l * l)) * Float64(k_m * k_m)) * t));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t) * 2.0) / Float64(k_m * k_m)) / Float64(k_m * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1.1e-132)
		tmp = 2.0 / ((((k_m * k_m) / (l * l)) * (k_m * k_m)) * t);
	else
		tmp = ((((l * l) / t) * 2.0) / (k_m * k_m)) / (k_m * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1.1e-132], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.09999999999999995e-132

   1. Initial program 31.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 66.2

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

   5. Applied rewrites 66.2%

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

   7. Applied rewrites 63.3%

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

   if 1.09999999999999995e-132 < l

   1. Initial program 39.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 59.9%

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

   13. Applied rewrites 60.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot 2}{k \cdot k}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 62.7

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

5. Applied rewrites 62.7%

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

7. Applied rewrites 60.9%

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

9. Applied rewrites 60.9%

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

10. Final simplification 60.9%

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

Alternative 11: 64.5% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 62.7

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

5. Applied rewrites 62.7%

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

7. Applied rewrites 60.9%

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

8. Final simplification 60.9%

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

Alternative 12: 31.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.3333333333333333}{k\_m} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ -0.3333333333333333 k_m) (/ (/ (* l l) t) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.3333333333333333 / k_m) * (((l * l) / 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 = ((-0.3333333333333333d0) / k_m) * (((l * l) / t) / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (-0.3333333333333333 / k_m) * (((l * l) / t) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (-0.3333333333333333 / k_m) * (((l * l) / t) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(-0.3333333333333333 / k_m) * Float64(Float64(Float64(l * l) / t) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (-0.3333333333333333 / k_m) * (((l * l) / t) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(-0.3333333333333333 / k$95$m), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 91.3%

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

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

   2. div-add N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}}}{{k}^{4}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} + \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

   6. associate-*r/ N/A

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

   7. associate-/l/ N/A

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

   8. *-commutative N/A

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

   9. div-add-rev N/A

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

   10. lower-/.f64 N/A

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

8. Applied rewrites 35.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2 \cdot \ell, \ell, \left(-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(k \cdot k\right)\right)}{{k}^{4} \cdot t}} \]

9. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 29.6%

    \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]

12. Final simplification 29.6%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k} \]
13. Add Preprocessing

Alternative 13: 21.4% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 1.0d0 / (t / ((-0.11666666666666667d0) * (l * l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 91.3%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

9. Taylor expanded in k around inf

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

11. Applied rewrites 20.3%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
12. Step-by-step derivation

13. Applied rewrites 20.3%

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

Alternative 14: 21.3% accurate, 21.0× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (-0.11666666666666667 * (l * l)) / 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 = ((-0.11666666666666667d0) * (l * l)) / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 91.3%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

9. Taylor expanded in k around inf

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

11. Applied rewrites 20.3%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
12. Step-by-step derivation

13. Applied rewrites 20.3%

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

Alternative 15: 21.3% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 91.3%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

9. Taylor expanded in k around inf

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

11. Applied rewrites 20.3%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
12. Step-by-step derivation

13. Applied rewrites 20.3%

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

14. Final simplification 20.3%

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

Alternative 16: 19.0% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 91.3%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k, k, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

9. Taylor expanded in k around inf

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

11. Applied rewrites 20.3%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
12. Step-by-step derivation

13. Applied rewrites 17.8%

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

14. Final simplification 17.8%

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

Reproduce

herbie shell --seed 2024298 
(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))))