Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 93.3%

Time: 18.6s

Alternatives: 6

Speedup: 3.8×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 35.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-96} \lor \neg \left(k \leq 9.6 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2} \cdot \frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= k 2.55e-96) (not (<= k 9.6e+124)))
   (/ 2.0 (* (pow (* (/ k l) (sin k)) 2.0) (/ t (cos k))))
   (/ 2.0 (/ (* (/ (* t (pow k 2.0)) (cos k)) (/ (pow (sin k) 2.0) l)) l))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((k <= 2.55e-96) || !(k <= 9.6e+124)) {
		tmp = 2.0 / (pow(((k / l) * sin(k)), 2.0) * (t / cos(k)));
	} else {
		tmp = 2.0 / ((((t * pow(k, 2.0)) / cos(k)) * (pow(sin(k), 2.0) / l)) / l);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= 2.55d-96) .or. (.not. (k <= 9.6d+124))) then
        tmp = 2.0d0 / ((((k / l) * sin(k)) ** 2.0d0) * (t / cos(k)))
    else
        tmp = 2.0d0 / ((((t * (k ** 2.0d0)) / cos(k)) * ((sin(k) ** 2.0d0) / l)) / l)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 2.55e-96) || !(k <= 9.6e+124)) {
		tmp = 2.0 / (Math.pow(((k / l) * Math.sin(k)), 2.0) * (t / Math.cos(k)));
	} else {
		tmp = 2.0 / ((((t * Math.pow(k, 2.0)) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (k <= 2.55e-96) or not (k <= 9.6e+124):
		tmp = 2.0 / (math.pow(((k / l) * math.sin(k)), 2.0) * (t / math.cos(k)))
	else:
		tmp = 2.0 / ((((t * math.pow(k, 2.0)) / math.cos(k)) * (math.pow(math.sin(k), 2.0) / l)) / l)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((k <= 2.55e-96) || !(k <= 9.6e+124))
		tmp = Float64(2.0 / Float64((Float64(Float64(k / l) * sin(k)) ^ 2.0) * Float64(t / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (k ^ 2.0)) / cos(k)) * Float64((sin(k) ^ 2.0) / l)) / l));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= 2.55e-96) || ~((k <= 9.6e+124)))
		tmp = 2.0 / ((((k / l) * sin(k)) ^ 2.0) * (t / cos(k)));
	else
		tmp = 2.0 / ((((t * (k ^ 2.0)) / cos(k)) * ((sin(k) ^ 2.0) / l)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[k, 2.55e-96], N[Not[LessEqual[k, 9.6e+124]], $MachinePrecision]], N[(2.0 / N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.54999999999999986e-96 or 9.60000000000000026e124 < k

   1. Initial program 39.3%

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

   3. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 46.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 46.3%

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

      3. *-commutative 46.3%

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

      4. associate-*r* 46.3%

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

      5. *-commutative 46.3%

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

      6. add-sqr-sqrt 16.3%

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

      7. pow2 16.3%

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

   5. Applied egg-rr 16.7%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 34.9%

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

      1. associate-/l* 35.3%

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

   8. Simplified 35.3%

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

      1. unpow-prod-down 33.0%

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

      2. associate-/r/ 33.0%

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

      3. pow2 33.0%

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

      4. add-sqr-sqrt 93.1%

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

   10. Applied egg-rr 93.1%

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

   if 2.54999999999999986e-96 < k < 9.60000000000000026e124

   1. Initial program 31.8%

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

   3. Simplified 35.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 35.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 35.9%

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

      3. *-commutative 35.9%

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

      4. associate-*r* 35.9%

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

      5. *-commutative 35.9%

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

      6. *-commutative 35.9%

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

      7. associate-/r* 37.8%

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

      8. associate-*r/ 39.6%

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

   5. Applied egg-rr 39.6%

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

   6. Taylor expanded in k around inf 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. times-frac 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-96} \lor \neg \left(k \leq 9.6 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2} \cdot \frac{t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \end{array} \]

Alternative 2: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-262}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t 9.5e-262)
   (/ 2.0 (/ (/ (/ (* t (pow k 4.0)) l) (cos k)) l))
   (/ 2.0 (pow (* (/ k (/ l (sin k))) (sqrt t)) 2.0))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e-262) {
		tmp = 2.0 / ((((t * pow(k, 4.0)) / l) / cos(k)) / l);
	} else {
		tmp = 2.0 / pow(((k / (l / sin(k))) * sqrt(t)), 2.0);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 9.5d-262) then
        tmp = 2.0d0 / ((((t * (k ** 4.0d0)) / l) / cos(k)) / l)
    else
        tmp = 2.0d0 / (((k / (l / sin(k))) * sqrt(t)) ** 2.0d0)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e-262) {
		tmp = 2.0 / ((((t * Math.pow(k, 4.0)) / l) / Math.cos(k)) / l);
	} else {
		tmp = 2.0 / Math.pow(((k / (l / Math.sin(k))) * Math.sqrt(t)), 2.0);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= 9.5e-262:
		tmp = 2.0 / ((((t * math.pow(k, 4.0)) / l) / math.cos(k)) / l)
	else:
		tmp = 2.0 / math.pow(((k / (l / math.sin(k))) * math.sqrt(t)), 2.0)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= 9.5e-262)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (k ^ 4.0)) / l) / cos(k)) / l));
	else
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l / sin(k))) * sqrt(t)) ^ 2.0));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 9.5e-262)
		tmp = 2.0 / ((((t * (k ^ 4.0)) / l) / cos(k)) / l);
	else
		tmp = 2.0 / (((k / (l / sin(k))) * sqrt(t)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, 9.5e-262], N[(2.0 / N[(N[(N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999999e-262

   1. Initial program 36.8%

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

   3. Simplified 41.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 41.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 41.3%

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

      3. *-commutative 41.3%

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

      4. associate-*r* 41.3%

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

      5. *-commutative 41.3%

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

      6. *-commutative 41.3%

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

      7. associate-/r* 43.7%

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

      8. associate-*r/ 44.0%

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

   5. Applied egg-rr 44.0%

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

   6. Taylor expanded in k around inf 83.5%

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

      1. times-frac 85.6%

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

   8. Simplified 85.6%

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

      1. associate-*r/ 85.6%

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

   10. Applied egg-rr 85.6%

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

   11. Taylor expanded in k around 0 69.4%

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

   if 9.4999999999999999e-262 < t

   1. Initial program 39.3%

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

   3. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l* 48.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

      2. +-rgt-identity 48.6%

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

      3. *-commutative 48.6%

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

      4. associate-*r* 48.6%

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

      5. *-commutative 48.6%

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

      6. add-sqr-sqrt 32.2%

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

      7. pow2 32.2%

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

   5. Applied egg-rr 51.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

   6. Taylor expanded in k around inf 65.8%

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

      1. associate-/l* 66.8%

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

   8. Simplified 66.8%

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

   9. Taylor expanded in k around 0 76.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-262}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 3: 92.4% accurate, 1.3× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (pow (* (/ k l) (sin k)) 2.0) (/ t (cos k)))))

C

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

Fortran

NOTE: k should be positive before calling this function
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 / ((((k / l) * sin(k)) ** 2.0d0) * (t / cos(k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*l* 44.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

   2. +-rgt-identity 44.1%

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

   3. *-commutative 44.1%

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

   4. associate-*r* 44.1%

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

   5. *-commutative 44.1%

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

   6. add-sqr-sqrt 17.7%

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

   7. pow2 17.7%

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

5. Applied egg-rr 20.2%

   \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

6. Taylor expanded in k around inf 39.3%

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

   1. associate-/l* 39.7%

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

8. Simplified 39.7%

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

   1. unpow-prod-down 36.0%

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

   2. associate-/r/ 36.0%

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

   3. pow2 36.0%

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

   4. add-sqr-sqrt 90.8%

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

10. Applied egg-rr 90.8%

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

11. Final simplification 90.8%

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

Alternative 4: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{\frac{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\cos k}}{\ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (/ (/ (/ (* t (pow k 4.0)) l) (cos k)) l)))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / ((((t * pow(k, 4.0)) / l) / cos(k)) / l);
}

Fortran

NOTE: k should be positive before calling this function
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 * (k ** 4.0d0)) / l) / cos(k)) / l)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 / ((((t * Math.pow(k, 4.0)) / l) / Math.cos(k)) / l);
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((((t * math.pow(k, 4.0)) / l) / math.cos(k)) / l)

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64(t * (k ^ 4.0)) / l) / cos(k)) / l))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 / ((((t * (k ^ 4.0)) / l) / cos(k)) / l);
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*l* 44.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

   2. +-rgt-identity 44.1%

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

   3. *-commutative 44.1%

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

   4. associate-*r* 44.1%

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

   5. *-commutative 44.1%

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

   6. *-commutative 44.1%

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

   7. associate-/r* 47.6%

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

   8. associate-*r/ 48.2%

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

5. Applied egg-rr 48.2%

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

6. Taylor expanded in k around inf 83.1%

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

   1. times-frac 85.5%

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

8. Simplified 85.5%

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

   1. associate-*r/ 85.5%

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

10. Applied egg-rr 85.5%

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

11. Taylor expanded in k around 0 69.4%

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

12. Final simplification 69.4%

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

Alternative 5: 65.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \ell \cdot \frac{2}{\frac{{k}^{4}}{\frac{\ell}{t}}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* l (/ 2.0 (/ (pow k 4.0) (/ l t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return l * (2.0 / (pow(k, 4.0) / (l / t)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (2.0d0 / ((k ** 4.0d0) / (l / t)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return l * (2.0 / (Math.pow(k, 4.0) / (l / t)));
}

Python

k = abs(k)
def code(t, l, k):
	return l * (2.0 / (math.pow(k, 4.0) / (l / t)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(l * Float64(2.0 / Float64((k ^ 4.0) / Float64(l / t))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = l * (2.0 / ((k ^ 4.0) / (l / t)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(l * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*l* 44.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

   2. +-rgt-identity 44.1%

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

   3. *-commutative 44.1%

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

   4. associate-*r* 44.1%

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

   5. *-commutative 44.1%

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

   6. *-commutative 44.1%

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

   7. associate-/r* 47.6%

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

   8. associate-*r/ 48.2%

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

5. Applied egg-rr 48.2%

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

6. Taylor expanded in k around 0 67.6%

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

   1. associate-/r/ 67.6%

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

   2. associate-/l* 66.3%

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

8. Applied egg-rr 66.3%

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

9. Final simplification 66.3%

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

Alternative 6: 67.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{2}{\frac{t \cdot \frac{{k}^{4}}{\ell}}{\ell}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (/ 2.0 (/ (* t (/ (pow k 4.0) l)) l)))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 / ((t * (pow(k, 4.0) / l)) / l);
}

Fortran

NOTE: k should be positive before calling this function
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 * ((k ** 4.0d0) / l)) / l)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 / ((t * (Math.pow(k, 4.0) / l)) / l);
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 / ((t * (math.pow(k, 4.0) / l)) / l)

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 / Float64(Float64(t * Float64((k ^ 4.0) / l)) / l))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 / ((t * ((k ^ 4.0) / l)) / l);
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*l* 44.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)\right)}} \]

   2. +-rgt-identity 44.1%

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

   3. *-commutative 44.1%

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

   4. associate-*r* 44.1%

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

   5. *-commutative 44.1%

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

   6. *-commutative 44.1%

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

   7. associate-/r* 47.6%

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

   8. associate-*r/ 48.2%

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

5. Applied egg-rr 48.2%

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

6. Taylor expanded in k around inf 83.1%

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

   1. times-frac 85.5%

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

8. Simplified 85.5%

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

9. Taylor expanded in k around 0 67.6%

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

    1. associate-*l/ 68.0%

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

11. Simplified 68.0%

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

12. Final simplification 68.0%

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

Reproduce

herbie shell --seed 2023333 
(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))))