Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 97.4%

Time: 13.8s

Alternatives: 24

Speedup: 11.0×

• 39.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\sin k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k\_m}{\cos k\_m}}{\ell} \cdot \left({\sin k\_m}^{2} \cdot t\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e-103)
   (/ 2.0 (* (sin k_m) (* (sin k_m) (* (/ k_m l) (* (/ k_m l) t)))))
   (/
    2.0
    (* (* (/ (/ k_m (cos k_m)) l) (* (pow (sin k_m) 2.0) t)) (/ k_m l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-103) {
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = 2.0 / ((((k_m / cos(k_m)) / l) * (pow(sin(k_m), 2.0) * t)) * (k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.5d-103) then
        tmp = 2.0d0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
    else
        tmp = 2.0d0 / ((((k_m / cos(k_m)) / l) * ((sin(k_m) ** 2.0d0) * t)) * (k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-103) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = 2.0 / ((((k_m / Math.cos(k_m)) / l) * (Math.pow(Math.sin(k_m), 2.0) * t)) * (k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.5e-103:
		tmp = 2.0 / (math.sin(k_m) * (math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
	else:
		tmp = 2.0 / ((((k_m / math.cos(k_m)) / l) * (math.pow(math.sin(k_m), 2.0) * t)) * (k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e-103)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(sin(k_m) * Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / cos(k_m)) / l) * Float64((sin(k_m) ^ 2.0) * t)) * Float64(k_m / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.5e-103)
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	else
		tmp = 2.0 / ((((k_m / cos(k_m)) / l) * ((sin(k_m) ^ 2.0) * t)) * (k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.5e-103], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.5e-103

   1. Initial program 34.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 87.9%

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

   7. Applied rewrites 95.1%

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.2%

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

   if 7.5e-103 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 98.0%

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

   9. Applied rewrites 98.0%

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

Alternative 2: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\sin k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left({\sin k\_m}^{2} \cdot \frac{k\_m}{\ell}\right)\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6e-103)
   (/ 2.0 (* (sin k_m) (* (sin k_m) (* (/ k_m l) (* (/ k_m l) t)))))
   (/
    2.0
    (* (* t (* (pow (sin k_m) 2.0) (/ k_m l))) (/ (/ k_m (cos k_m)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6e-103) {
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = 2.0 / ((t * (pow(sin(k_m), 2.0) * (k_m / l))) * ((k_m / cos(k_m)) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6d-103) then
        tmp = 2.0d0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
    else
        tmp = 2.0d0 / ((t * ((sin(k_m) ** 2.0d0) * (k_m / l))) * ((k_m / cos(k_m)) / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6e-103) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = 2.0 / ((t * (Math.pow(Math.sin(k_m), 2.0) * (k_m / l))) * ((k_m / Math.cos(k_m)) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6e-103:
		tmp = 2.0 / (math.sin(k_m) * (math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
	else:
		tmp = 2.0 / ((t * (math.pow(math.sin(k_m), 2.0) * (k_m / l))) * ((k_m / math.cos(k_m)) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6e-103)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(sin(k_m) * Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64((sin(k_m) ^ 2.0) * Float64(k_m / l))) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 6e-103)
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	else
		tmp = 2.0 / ((t * ((sin(k_m) ^ 2.0) * (k_m / l))) * ((k_m / cos(k_m)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6e-103], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e-103

   1. Initial program 34.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 87.9%

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

   7. Applied rewrites 95.1%

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.2%

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

   if 6e-103 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 98.0%

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

Alternative 3: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 89.6%

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

7. Applied rewrites 96.1%

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

9. Applied rewrites 98.4%

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

Alternative 4: 93.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\sin k\_m}^{2} \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot k\_m}{\cos k\_m \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(-t\right) \cdot \sin k\_m}{{\left(\frac{\ell}{k\_m}\right)}^{2}} \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.5e-120)
   (/ 2.0 (/ (* (* (* (pow (sin k_m) 2.0) t) (/ k_m l)) k_m) (* (cos k_m) l)))
   (/ -2.0 (* (/ (* (- t) (sin k_m)) (pow (/ l k_m) 2.0)) (tan k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.5e-120) {
		tmp = 2.0 / ((((pow(sin(k_m), 2.0) * t) * (k_m / l)) * k_m) / (cos(k_m) * l));
	} else {
		tmp = -2.0 / (((-t * sin(k_m)) / pow((l / k_m), 2.0)) * tan(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 (t <= 3.5d-120) then
        tmp = 2.0d0 / (((((sin(k_m) ** 2.0d0) * t) * (k_m / l)) * k_m) / (cos(k_m) * l))
    else
        tmp = (-2.0d0) / (((-t * sin(k_m)) / ((l / k_m) ** 2.0d0)) * tan(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 (t <= 3.5e-120) {
		tmp = 2.0 / ((((Math.pow(Math.sin(k_m), 2.0) * t) * (k_m / l)) * k_m) / (Math.cos(k_m) * l));
	} else {
		tmp = -2.0 / (((-t * Math.sin(k_m)) / Math.pow((l / k_m), 2.0)) * Math.tan(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3.5e-120:
		tmp = 2.0 / ((((math.pow(math.sin(k_m), 2.0) * t) * (k_m / l)) * k_m) / (math.cos(k_m) * l))
	else:
		tmp = -2.0 / (((-t * math.sin(k_m)) / math.pow((l / k_m), 2.0)) * math.tan(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.5e-120)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * Float64(k_m / l)) * k_m) / Float64(cos(k_m) * l)));
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(-t) * sin(k_m)) / (Float64(l / k_m) ^ 2.0)) * tan(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3.5e-120)
		tmp = 2.0 / (((((sin(k_m) ^ 2.0) * t) * (k_m / l)) * k_m) / (cos(k_m) * l));
	else
		tmp = -2.0 / (((-t * sin(k_m)) / ((l / k_m) ^ 2.0)) * tan(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.5e-120], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(N[((-t) * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e-120

   1. Initial program 30.0%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   7. Applied rewrites 95.6%

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

   9. Applied rewrites 91.3%

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

   if 3.5e-120 < t

   1. Initial program 40.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 97.5%

      \[\leadsto \frac{-2}{\frac{\left(\left(-t\right) \cdot \sin k\right) \cdot 1}{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}} \cdot \tan k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\sin k}^{2} \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(-t\right) \cdot \sin k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \frac{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(-t\right) \cdot \sin k\_m}{{\left(\frac{\ell}{k\_m}\right)}^{2}} \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.3e-154)
   (/ 2.0 (* (/ k_m (* (cos k_m) l)) (/ (* (* (pow (sin k_m) 2.0) t) k_m) l)))
   (/ -2.0 (* (/ (* (- t) (sin k_m)) (pow (/ l k_m) 2.0)) (tan k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.3e-154) {
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * (((pow(sin(k_m), 2.0) * t) * k_m) / l));
	} else {
		tmp = -2.0 / (((-t * sin(k_m)) / pow((l / k_m), 2.0)) * tan(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 (t <= 4.3d-154) then
        tmp = 2.0d0 / ((k_m / (cos(k_m) * l)) * ((((sin(k_m) ** 2.0d0) * t) * k_m) / l))
    else
        tmp = (-2.0d0) / (((-t * sin(k_m)) / ((l / k_m) ** 2.0d0)) * tan(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 (t <= 4.3e-154) {
		tmp = 2.0 / ((k_m / (Math.cos(k_m) * l)) * (((Math.pow(Math.sin(k_m), 2.0) * t) * k_m) / l));
	} else {
		tmp = -2.0 / (((-t * Math.sin(k_m)) / Math.pow((l / k_m), 2.0)) * Math.tan(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.3e-154:
		tmp = 2.0 / ((k_m / (math.cos(k_m) * l)) * (((math.pow(math.sin(k_m), 2.0) * t) * k_m) / l))
	else:
		tmp = -2.0 / (((-t * math.sin(k_m)) / math.pow((l / k_m), 2.0)) * math.tan(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.3e-154)
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m) / l)));
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(-t) * sin(k_m)) / (Float64(l / k_m) ^ 2.0)) * tan(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 4.3e-154)
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * ((((sin(k_m) ^ 2.0) * t) * k_m) / l));
	else
		tmp = -2.0 / (((-t * sin(k_m)) / ((l / k_m) ^ 2.0)) * tan(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.3e-154], N[(2.0 / N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(N[((-t) * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.29999999999999992e-154

   1. Initial program 28.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in t around 0

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

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

      7. times-frac N/A

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

      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. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 89.8

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

   8. Applied rewrites 89.8%

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

   if 4.29999999999999992e-154 < t

   1. Initial program 41.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.2

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 97.7%

      \[\leadsto \frac{-2}{\frac{\left(\left(-t\right) \cdot \sin k\right) \cdot 1}{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}} \cdot \tan k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k \cdot \ell} \cdot \frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(-t\right) \cdot \sin k}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-neg-in N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

4. Applied rewrites 46.4%

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

5. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 83.9

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

7. Applied rewrites 83.9%

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

9. Applied rewrites 95.2%

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

10. Final simplification 95.2%

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

Alternative 7: 82.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-282}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\sin k\_m \cdot \left(\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(\sin k\_m \cdot t\right) \cdot \left(\left(-k\_m\right) \cdot k\_m\right)}{\ell \cdot \ell} \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 4e-282)
   (/ 2.0 (* (sin k_m) (* (sin k_m) (* (/ k_m l) (* (/ k_m l) t)))))
   (/ -2.0 (* (/ (* (* (sin k_m) t) (* (- k_m) k_m)) (* l l)) (tan k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e-282) {
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = -2.0 / ((((sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * tan(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 * l) <= 4d-282) then
        tmp = 2.0d0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
    else
        tmp = (-2.0d0) / ((((sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * tan(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 * l) <= 4e-282) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = -2.0 / ((((Math.sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * Math.tan(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 4e-282:
		tmp = 2.0 / (math.sin(k_m) * (math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
	else:
		tmp = -2.0 / ((((math.sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * math.tan(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 4e-282)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(sin(k_m) * Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t)))));
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(sin(k_m) * t) * Float64(Float64(-k_m) * k_m)) / Float64(l * l)) * tan(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 4e-282)
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	else
		tmp = -2.0 / ((((sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * tan(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e-282], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * N[((-k$95$m) * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.0000000000000001e-282

   1. Initial program 22.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 90.1%

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 95.3%

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

   if 4.0000000000000001e-282 < (*.f64 l l)

   1. Initial program 37.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

   9. Applied rewrites 78.2%

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

Alternative 8: 95.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{-2}{\left(\left(\left(-t\right) \cdot \sin k\_m\right) \cdot \frac{\frac{k\_m}{\ell} \cdot k\_m}{\ell}\right) \cdot \tan k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\left(\left(\sin k\_m \cdot \frac{t}{\ell}\right) \cdot k\_m\right) \cdot \frac{-k\_m}{\ell}\right) \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.6e-128)
   (/ -2.0 (* (* (* (- t) (sin k_m)) (/ (* (/ k_m l) k_m) l)) (tan k_m)))
   (/ -2.0 (* (* (* (* (sin k_m) (/ t l)) k_m) (/ (- k_m) l)) (tan k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6e-128) {
		tmp = -2.0 / (((-t * sin(k_m)) * (((k_m / l) * k_m) / l)) * tan(k_m));
	} else {
		tmp = -2.0 / ((((sin(k_m) * (t / l)) * k_m) * (-k_m / l)) * tan(k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.6d-128) then
        tmp = (-2.0d0) / (((-t * sin(k_m)) * (((k_m / l) * k_m) / l)) * tan(k_m))
    else
        tmp = (-2.0d0) / ((((sin(k_m) * (t / l)) * k_m) * (-k_m / l)) * tan(k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6e-128) {
		tmp = -2.0 / (((-t * Math.sin(k_m)) * (((k_m / l) * k_m) / l)) * Math.tan(k_m));
	} else {
		tmp = -2.0 / ((((Math.sin(k_m) * (t / l)) * k_m) * (-k_m / l)) * Math.tan(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.6e-128:
		tmp = -2.0 / (((-t * math.sin(k_m)) * (((k_m / l) * k_m) / l)) * math.tan(k_m))
	else:
		tmp = -2.0 / ((((math.sin(k_m) * (t / l)) * k_m) * (-k_m / l)) * math.tan(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.6e-128)
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(-t) * sin(k_m)) * Float64(Float64(Float64(k_m / l) * k_m) / l)) * tan(k_m)));
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(sin(k_m) * Float64(t / l)) * k_m) * Float64(Float64(-k_m) / l)) * tan(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.6e-128)
		tmp = -2.0 / (((-t * sin(k_m)) * (((k_m / l) * k_m) / l)) * tan(k_m));
	else
		tmp = -2.0 / ((((sin(k_m) * (t / l)) * k_m) * (-k_m / l)) * tan(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.6e-128], N[(-2.0 / N[(N[(N[((-t) * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[((-k$95$m) / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.6000000000000005e-128

   1. Initial program 34.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

   9. Applied rewrites 95.7%

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

   if 7.6000000000000005e-128 < k

   1. Initial program 30.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 90.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{-2}{\left(\left(\left(-t\right) \cdot \sin k\right) \cdot \frac{\frac{k}{\ell} \cdot k}{\ell}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \frac{-k}{\ell}\right) \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.9% accurate, 1.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4e-10)
   (/ 2.0 (* (sin k_m) (* (sin k_m) (* (/ k_m l) (* (/ k_m l) t)))))
   (/ -2.0 (* (* (* (* (sin k_m) (tan k_m)) (/ t l)) k_m) (/ (- k_m) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-10) {
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = -2.0 / ((((sin(k_m) * tan(k_m)) * (t / l)) * k_m) * (-k_m / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4d-10) then
        tmp = 2.0d0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
    else
        tmp = (-2.0d0) / ((((sin(k_m) * tan(k_m)) * (t / l)) * k_m) * (-k_m / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-10) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	} else {
		tmp = -2.0 / ((((Math.sin(k_m) * Math.tan(k_m)) * (t / l)) * k_m) * (-k_m / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4e-10:
		tmp = 2.0 / (math.sin(k_m) * (math.sin(k_m) * ((k_m / l) * ((k_m / l) * t))))
	else:
		tmp = -2.0 / ((((math.sin(k_m) * math.tan(k_m)) * (t / l)) * k_m) * (-k_m / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-10)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(sin(k_m) * Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t)))));
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(t / l)) * k_m) * Float64(Float64(-k_m) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4e-10)
		tmp = 2.0 / (sin(k_m) * (sin(k_m) * ((k_m / l) * ((k_m / l) * t))));
	else
		tmp = -2.0 / ((((sin(k_m) * tan(k_m)) * (t / l)) * k_m) * (-k_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4e-10], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[((-k$95$m) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000015e-10

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 88.4%

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

   7. Applied rewrites 95.2%

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

   9. Applied rewrites 98.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.5%

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

   if 4.00000000000000015e-10 < k

   1. Initial program 34.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 88.9%

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

4. Final simplification 84.2%

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

Alternative 10: 94.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.6 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{k\_m \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(-k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.6e-128)
   (pow (/ (* k_m t) (/ (* 2.0 (pow (/ l k_m) 2.0)) k_m)) -1.0)
   (/ -2.0 (* (- k_m) (* (/ k_m l) (* (* (sin k_m) (tan k_m)) (/ t l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.6e-128) {
		tmp = pow(((k_m * t) / ((2.0 * pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / (-k_m * ((k_m / l) * ((sin(k_m) * tan(k_m)) * (t / 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 <= 7.6d-128) then
        tmp = ((k_m * t) / ((2.0d0 * ((l / k_m) ** 2.0d0)) / k_m)) ** (-1.0d0)
    else
        tmp = (-2.0d0) / (-k_m * ((k_m / l) * ((sin(k_m) * tan(k_m)) * (t / 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 <= 7.6e-128) {
		tmp = Math.pow(((k_m * t) / ((2.0 * Math.pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / (-k_m * ((k_m / l) * ((Math.sin(k_m) * Math.tan(k_m)) * (t / l))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.6e-128:
		tmp = math.pow(((k_m * t) / ((2.0 * math.pow((l / k_m), 2.0)) / k_m)), -1.0)
	else:
		tmp = -2.0 / (-k_m * ((k_m / l) * ((math.sin(k_m) * math.tan(k_m)) * (t / l))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.6e-128)
		tmp = Float64(Float64(k_m * t) / Float64(Float64(2.0 * (Float64(l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = Float64(-2.0 / Float64(Float64(-k_m) * Float64(Float64(k_m / l) * Float64(Float64(sin(k_m) * tan(k_m)) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.6e-128)
		tmp = ((k_m * t) / ((2.0 * ((l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = -2.0 / (-k_m * ((k_m / l) * ((sin(k_m) * tan(k_m)) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.6e-128], N[Power[N[(N[(k$95$m * t), $MachinePrecision] / N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(-2.0 / N[((-k$95$m) * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.6000000000000005e-128

   1. Initial program 34.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 72.0%

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

   11. Applied rewrites 78.1%

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

   if 7.6000000000000005e-128 < k

   1. Initial program 30.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 81.4%

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

   9. Applied rewrites 90.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{k \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(-k\right) \cdot \left(\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-282}:\\ \;\;\;\;{\left(\frac{k\_m \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(\sin k\_m \cdot t\right) \cdot \left(\left(-k\_m\right) \cdot k\_m\right)}{\ell \cdot \ell} \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 4e-282)
   (pow (/ (* k_m t) (/ (* 2.0 (pow (/ l k_m) 2.0)) k_m)) -1.0)
   (/ -2.0 (* (/ (* (* (sin k_m) t) (* (- k_m) k_m)) (* l l)) (tan k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e-282) {
		tmp = pow(((k_m * t) / ((2.0 * pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / ((((sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * tan(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 * l) <= 4d-282) then
        tmp = ((k_m * t) / ((2.0d0 * ((l / k_m) ** 2.0d0)) / k_m)) ** (-1.0d0)
    else
        tmp = (-2.0d0) / ((((sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * tan(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 * l) <= 4e-282) {
		tmp = Math.pow(((k_m * t) / ((2.0 * Math.pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / ((((Math.sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * Math.tan(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 4e-282:
		tmp = math.pow(((k_m * t) / ((2.0 * math.pow((l / k_m), 2.0)) / k_m)), -1.0)
	else:
		tmp = -2.0 / ((((math.sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * math.tan(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 4e-282)
		tmp = Float64(Float64(k_m * t) / Float64(Float64(2.0 * (Float64(l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(sin(k_m) * t) * Float64(Float64(-k_m) * k_m)) / Float64(l * l)) * tan(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 4e-282)
		tmp = ((k_m * t) / ((2.0 * ((l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = -2.0 / ((((sin(k_m) * t) * (-k_m * k_m)) / (l * l)) * tan(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e-282], N[Power[N[(N[(k$95$m * t), $MachinePrecision] / N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(-2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * N[((-k$95$m) * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.0000000000000001e-282

   1. Initial program 22.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 84.0%

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

   9. Applied rewrites 83.1%

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

   11. Applied rewrites 95.1%

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

   if 4.0000000000000001e-282 < (*.f64 l l)

   1. Initial program 37.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 43.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

   9. Applied rewrites 78.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-282}:\\ \;\;\;\;{\left(\frac{k \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(\sin k \cdot t\right) \cdot \left(\left(-k\right) \cdot k\right)}{\ell \cdot \ell} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;{\left(\frac{k\_m \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\left(k\_m \cdot \frac{-t}{\ell}\right) \cdot \frac{k\_m \cdot k\_m}{\ell}\right) \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 2.5e+85)
   (pow (/ (* k_m t) (/ (* 2.0 (pow (/ l k_m) 2.0)) k_m)) -1.0)
   (/ -2.0 (* (* (* k_m (/ (- t) l)) (/ (* k_m k_m) l)) (tan k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2.5e+85) {
		tmp = pow(((k_m * t) / ((2.0 * pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / (((k_m * (-t / l)) * ((k_m * k_m) / l)) * tan(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 * l) <= 2.5d+85) then
        tmp = ((k_m * t) / ((2.0d0 * ((l / k_m) ** 2.0d0)) / k_m)) ** (-1.0d0)
    else
        tmp = (-2.0d0) / (((k_m * (-t / l)) * ((k_m * k_m) / l)) * tan(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 * l) <= 2.5e+85) {
		tmp = Math.pow(((k_m * t) / ((2.0 * Math.pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / (((k_m * (-t / l)) * ((k_m * k_m) / l)) * Math.tan(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (l * l) <= 2.5e+85:
		tmp = math.pow(((k_m * t) / ((2.0 * math.pow((l / k_m), 2.0)) / k_m)), -1.0)
	else:
		tmp = -2.0 / (((k_m * (-t / l)) * ((k_m * k_m) / l)) * math.tan(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2.5e+85)
		tmp = Float64(Float64(k_m * t) / Float64(Float64(2.0 * (Float64(l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(k_m * Float64(Float64(-t) / l)) * Float64(Float64(k_m * k_m) / l)) * tan(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 2.5e+85)
		tmp = ((k_m * t) / ((2.0 * ((l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = -2.0 / (((k_m * (-t / l)) * ((k_m * k_m) / l)) * tan(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 2.5e+85], N[Power[N[(N[(k$95$m * t), $MachinePrecision] / N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(-2.0 / N[(N[(N[(k$95$m * N[((-t) / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.5e85

   1. Initial program 28.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 73.8

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 79.9%

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

   11. Applied rewrites 86.9%

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

   if 2.5e85 < (*.f64 l l)

   1. Initial program 39.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 58.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2.5 \cdot 10^{+85}:\\ \;\;\;\;{\left(\frac{k \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\left(k \cdot \frac{-t}{\ell}\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;{\left(\frac{k\_m \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\cos k\_m \cdot \ell} \cdot \frac{\left(-k\_m\right) \cdot k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.6e-127)
   (pow (/ (* k_m t) (/ (* 2.0 (pow (/ l k_m) 2.0)) k_m)) -1.0)
   (/ -2.0 (* (/ (* (* k_m k_m) t) (* (cos k_m) l)) (/ (* (- k_m) k_m) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-127) {
		tmp = pow(((k_m * t) / ((2.0 * pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / ((((k_m * k_m) * t) / (cos(k_m) * l)) * ((-k_m * k_m) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.6d-127) then
        tmp = ((k_m * t) / ((2.0d0 * ((l / k_m) ** 2.0d0)) / k_m)) ** (-1.0d0)
    else
        tmp = (-2.0d0) / ((((k_m * k_m) * t) / (cos(k_m) * l)) * ((-k_m * k_m) / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-127) {
		tmp = Math.pow(((k_m * t) / ((2.0 * Math.pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = -2.0 / ((((k_m * k_m) * t) / (Math.cos(k_m) * l)) * ((-k_m * k_m) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.6e-127:
		tmp = math.pow(((k_m * t) / ((2.0 * math.pow((l / k_m), 2.0)) / k_m)), -1.0)
	else:
		tmp = -2.0 / ((((k_m * k_m) * t) / (math.cos(k_m) * l)) * ((-k_m * k_m) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6e-127)
		tmp = Float64(Float64(k_m * t) / Float64(Float64(2.0 * (Float64(l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = Float64(-2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / Float64(cos(k_m) * l)) * Float64(Float64(Float64(-k_m) * k_m) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.6e-127)
		tmp = ((k_m * t) / ((2.0 * ((l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = -2.0 / ((((k_m * k_m) * t) / (cos(k_m) * l)) * ((-k_m * k_m) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.6e-127], N[Power[N[(N[(k$95$m * t), $MachinePrecision] / N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(-2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[((-k$95$m) * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.59999999999999961e-127

   1. Initial program 34.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 71.6%

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

   11. Applied rewrites 78.3%

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

   if 6.59999999999999961e-127 < k

   1. Initial program 31.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 40.4%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 68.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;{\left(\frac{k \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k \cdot \ell} \cdot \frac{\left(-k\right) \cdot k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{k\_m \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot 2}{t} \cdot {k\_m}^{-2}\right) \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.2e-128)
   (pow (/ (* k_m t) (/ (* 2.0 (pow (/ l k_m) 2.0)) k_m)) -1.0)
   (* (* (/ (* l 2.0) t) (pow k_m -2.0)) (/ l (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e-128) {
		tmp = pow(((k_m * t) / ((2.0 * pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = (((l * 2.0) / t) * pow(k_m, -2.0)) * (l / (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.2d-128) then
        tmp = ((k_m * t) / ((2.0d0 * ((l / k_m) ** 2.0d0)) / k_m)) ** (-1.0d0)
    else
        tmp = (((l * 2.0d0) / t) * (k_m ** (-2.0d0))) * (l / (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e-128) {
		tmp = Math.pow(((k_m * t) / ((2.0 * Math.pow((l / k_m), 2.0)) / k_m)), -1.0);
	} else {
		tmp = (((l * 2.0) / t) * Math.pow(k_m, -2.0)) * (l / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.2e-128:
		tmp = math.pow(((k_m * t) / ((2.0 * math.pow((l / k_m), 2.0)) / k_m)), -1.0)
	else:
		tmp = (((l * 2.0) / t) * math.pow(k_m, -2.0)) * (l / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.2e-128)
		tmp = Float64(Float64(k_m * t) / Float64(Float64(2.0 * (Float64(l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / t) * (k_m ^ -2.0)) * Float64(l / 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 (k_m <= 6.2e-128)
		tmp = ((k_m * t) / ((2.0 * ((l / k_m) ^ 2.0)) / k_m)) ^ -1.0;
	else
		tmp = (((l * 2.0) / t) * (k_m ^ -2.0)) * (l / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.2e-128], N[Power[N[(N[(k$95$m * t), $MachinePrecision] / N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / t), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.20000000000000005e-128

   1. Initial program 34.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 72.0%

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

   11. Applied rewrites 78.1%

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

   if 6.20000000000000005e-128 < k

   1. Initial program 30.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 65.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{k \cdot t}{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot 2}{t} \cdot {k}^{-2}\right) \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k\_m}\right)}^{2}}{k\_m}}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot 2}{t} \cdot {k\_m}^{-2}\right) \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.2e-128)
   (/ (/ (* 2.0 (pow (/ l k_m) 2.0)) k_m) (* k_m t))
   (* (* (/ (* l 2.0) t) (pow k_m -2.0)) (/ l (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e-128) {
		tmp = ((2.0 * pow((l / k_m), 2.0)) / k_m) / (k_m * t);
	} else {
		tmp = (((l * 2.0) / t) * pow(k_m, -2.0)) * (l / (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.2d-128) then
        tmp = ((2.0d0 * ((l / k_m) ** 2.0d0)) / k_m) / (k_m * t)
    else
        tmp = (((l * 2.0d0) / t) * (k_m ** (-2.0d0))) * (l / (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e-128) {
		tmp = ((2.0 * Math.pow((l / k_m), 2.0)) / k_m) / (k_m * t);
	} else {
		tmp = (((l * 2.0) / t) * Math.pow(k_m, -2.0)) * (l / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6.2e-128:
		tmp = ((2.0 * math.pow((l / k_m), 2.0)) / k_m) / (k_m * t)
	else:
		tmp = (((l * 2.0) / t) * math.pow(k_m, -2.0)) * (l / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.2e-128)
		tmp = Float64(Float64(Float64(2.0 * (Float64(l / k_m) ^ 2.0)) / k_m) / Float64(k_m * t));
	else
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / t) * (k_m ^ -2.0)) * Float64(l / 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 (k_m <= 6.2e-128)
		tmp = ((2.0 * ((l / k_m) ^ 2.0)) / k_m) / (k_m * t);
	else
		tmp = (((l * 2.0) / t) * (k_m ^ -2.0)) * (l / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.2e-128], N[(N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / t), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.20000000000000005e-128

   1. Initial program 34.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 72.0%

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

   11. Applied rewrites 78.1%

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

   if 6.20000000000000005e-128 < k

   1. Initial program 30.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 65.6%

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

Alternative 16: 74.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot 2\right) \cdot {\left(t \cdot \left(\frac{k\_m}{\ell} \cdot k\_m\right)\right)}^{-1}}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* l 2.0) (pow (* t (* (/ k_m l) k_m)) -1.0)) (* k_m k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * 2.0) * pow((t * ((k_m / l) * k_m)), -1.0)) / (k_m * k_m);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * 2.0d0) * ((t * ((k_m / l) * k_m)) ** (-1.0d0))) / (k_m * k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * 2.0) * Math.pow((t * ((k_m / l) * k_m)), -1.0)) / (k_m * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * 2.0) * math.pow((t * ((k_m / l) * k_m)), -1.0)) / (k_m * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * 2.0) * (Float64(t * Float64(Float64(k_m / l) * k_m)) ^ -1.0)) / Float64(k_m * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * 2.0) * ((t * ((k_m / l) * k_m)) ^ -1.0)) / (k_m * k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * 2.0), $MachinePrecision] * N[Power[N[(t * N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 62.4%

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

9. Applied rewrites 72.3%

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

10. Final simplification 72.3%

    \[\leadsto \frac{\left(\ell \cdot 2\right) \cdot {\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)}^{-1}}{k \cdot k} \]
11. Add Preprocessing

Alternative 17: 74.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 62.4%

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

9. Applied rewrites 72.3%

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

Alternative 18: 73.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 62.4%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 72.3%

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

Alternative 19: 74.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 69.8%

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

11. Applied rewrites 71.9%

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

Alternative 20: 72.1% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 69.9%

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

10. Final simplification 69.9%

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

Alternative 21: 72.8% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 69.8%

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

11. Applied rewrites 69.8%

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

Alternative 22: 72.8% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 69.8%

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

Alternative 23: 72.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-pow.f64 65.4

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

5. Applied rewrites 65.4%

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 69.8%

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

11. Applied rewrites 69.0%

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

Alternative 24: 70.6% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * 2.0) * (l / (((k_m * k_m) * t) * (k_m * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * 2.0) * (l / (((k_m * k_m) * t) * (k_m * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * 2.0) * Float64(l / Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * 2.0) * (l / (((k_m * k_m) * t) * (k_m * k_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

   10. lower-pow.f64 65.4

       \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 65.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
6. Step-by-step derivation

7. Applied rewrites 68.8%

   \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\frac{\ell}{k \cdot k}}{\color{blue}{k \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 69.8%

   \[\leadsto \left(\ell \cdot 2\right) \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 68.0%

    \[\leadsto \left(\ell \cdot 2\right) \cdot \frac{-\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\left(-k\right) \cdot k\right)}} \]

12. Final simplification 68.0%

    \[\leadsto \left(\ell \cdot 2\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(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))))