Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 98.5%

Time: 13.0s

Alternatives: 14

Speedup: 11.0×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ k l) (* (* (/ (/ k (cos k)) l) t) (sin k))) (sin k))))

C

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

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 / (((k / l) * ((((k / cos(k)) / l) * t) * sin(k))) * sin(k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 90.2%

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

7. Applied rewrites 96.1%

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

9. Applied rewrites 97.6%

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

11. Applied rewrites 98.2%

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

Alternative 2: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\sin k}^{2} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot k}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.55e-100)
   (/
    2.0
    (*
     (*
      (/ (/ k (cos k)) l)
      (* (* (* (fma -0.3333333333333333 (* k k) 1.0) t) k) k))
     (/ k l)))
   (/ 2.0 (/ (* (* (pow (sin k) 2.0) (* t (/ k l))) k) (* (cos k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-100) {
		tmp = 2.0 / ((((k / cos(k)) / l) * (((fma(-0.3333333333333333, (k * k), 1.0) * t) * k) * k)) * (k / l));
	} else {
		tmp = 2.0 / (((pow(sin(k), 2.0) * (t * (k / l))) * k) / (cos(k) * l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.55e-100)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / cos(k)) / l) * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * t) * k) * k)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) * Float64(t * Float64(k / l))) * k) / Float64(cos(k) * l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.55e-100], N[(2.0 / N[(N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5499999999999999e-100

   1. Initial program 35.7%

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

   3. Taylor expanded in 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 89.9%

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 79.1%

       \[\leadsto \frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}} \]

   if 1.5499999999999999e-100 < k

   1. Initial program 30.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 90.9%

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

   7. Applied rewrites 91.0%

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

   9. Applied rewrites 94.5%

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

Alternative 3: 97.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ k (* (cos k) l)) (* (* t (sin k)) (sin k))) (/ k l))))

C

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

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 / (((k / (cos(k) * l)) * ((t * sin(k)) * sin(k))) * (k / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 90.2%

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

7. Applied rewrites 96.1%

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

9. Applied rewrites 97.6%

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

11. Applied rewrites 97.6%

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

Alternative 4: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.066:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.06 \cdot 10^{+212}:\\ \;\;\;\;\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(\left(t \cdot \sin k\right) \cdot \sin k\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.066)
   (/
    2.0
    (*
     (*
      (/ (/ k (cos k)) l)
      (* (* (* (fma -0.3333333333333333 (* k k) 1.0) t) k) k))
     (/ k l)))
   (if (<= k 1.06e+212)
     (*
      (* (* (cos k) l) l)
      (/ (/ (/ 2.0 (* (- 0.5 (* 0.5 (cos (+ k k)))) t)) k) k))
     (/ 2.0 (* (* (/ k l) (* (* t (sin k)) (sin k))) (/ k l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.066) {
		tmp = 2.0 / ((((k / cos(k)) / l) * (((fma(-0.3333333333333333, (k * k), 1.0) * t) * k) * k)) * (k / l));
	} else if (k <= 1.06e+212) {
		tmp = ((cos(k) * l) * l) * (((2.0 / ((0.5 - (0.5 * cos((k + k)))) * t)) / k) / k);
	} else {
		tmp = 2.0 / (((k / l) * ((t * sin(k)) * sin(k))) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.066)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / cos(k)) / l) * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * t) * k) * k)) * Float64(k / l)));
	elseif (k <= 1.06e+212)
		tmp = Float64(Float64(Float64(cos(k) * l) * l) * Float64(Float64(Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t)) / k) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(t * sin(k)) * sin(k))) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.066], N[(2.0 / N[(N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.06e+212], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.066000000000000003

   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 t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.5%

       \[\leadsto \frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}} \]

   if 0.066000000000000003 < k < 1.05999999999999995e212

   1. Initial program 27.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 74.9%

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

   if 1.05999999999999995e212 < k

   1. Initial program 42.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 80.9%

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

Alternative 5: 83.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{k}{\cos k}}{\ell}\\ \mathbf{if}\;k \leq 0.0027:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}{\ell} \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ k (cos k)) l)))
   (if (<= k 0.0027)
     (/
      2.0
      (*
       (* t_1 (* (* (* (fma -0.3333333333333333 (* k k) 1.0) t) k) k))
       (/ k l)))
     (/ 2.0 (* (/ (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t) k) l) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = (k / cos(k)) / l;
	double tmp;
	if (k <= 0.0027) {
		tmp = 2.0 / ((t_1 * (((fma(-0.3333333333333333, (k * k), 1.0) * t) * k) * k)) * (k / l));
	} else {
		tmp = 2.0 / (((((0.5 - (0.5 * cos((k + k)))) * t) * k) / l) * t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / cos(k)) / l)
	tmp = 0.0
	if (k <= 0.0027)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * t) * k) * k)) * Float64(k / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t) * k) / l) * t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k, 0.0027], N[(2.0 / N[(N[(t$95$1 * N[(N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0027000000000000001

   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 t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.5%

       \[\leadsto \frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}} \]

   if 0.0027000000000000001 < k

   1. Initial program 32.7%

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

   3. Taylor expanded in t around 0

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

      1. *-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 87.4%

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

   7. Applied rewrites 85.6%

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

Alternative 6: 77.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ k l) (* (* t (sin k)) (sin k))) (/ k l))))

C

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

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 / (((k / l) * ((t * sin(k)) * sin(k))) * (k / l))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(t * sin(k)) * sin(k))) * Float64(k / l)))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 90.2%

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

7. Applied rewrites 96.1%

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

9. Applied rewrites 97.6%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 79.7%

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

Alternative 7: 74.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+213}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot \frac{\frac{\frac{\frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, 0.6666666666666666, \frac{2}{t}\right)}{k}}{k}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.8e+213)
   (/
    2.0
    (*
     (*
      (/ (/ k (cos k)) l)
      (* (* (* (fma -0.3333333333333333 (* k k) 1.0) t) k) k))
     (/ k l)))
   (*
    (* (* (cos k) l) l)
    (/ (/ (/ (/ (fma (/ (* k k) t) 0.6666666666666666 (/ 2.0 t)) k) k) k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.8e+213) {
		tmp = 2.0 / ((((k / cos(k)) / l) * (((fma(-0.3333333333333333, (k * k), 1.0) * t) * k) * k)) * (k / l));
	} else {
		tmp = ((cos(k) * l) * l) * ((((fma(((k * k) / t), 0.6666666666666666, (2.0 / t)) / k) / k) / k) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.8e+213)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / cos(k)) / l) * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * t) * k) * k)) * Float64(k / l)));
	else
		tmp = Float64(Float64(Float64(cos(k) * l) * l) * Float64(Float64(Float64(Float64(fma(Float64(Float64(k * k) / t), 0.6666666666666666, Float64(2.0 / t)) / k) / k) / k) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.8e+213], N[(2.0 / N[(N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / t), $MachinePrecision] * 0.6666666666666666 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7999999999999999e213

   1. Initial program 34.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 90.5%

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 78.1%

       \[\leadsto \frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}} \]

   if 2.7999999999999999e213 < l

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 73.0%

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

Alternative 8: 75.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{k}{\cos k}}{\ell}\\ \mathbf{if}\;t \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ k (cos k)) l)))
   (if (<= t 1.5e-15)
     (/ 2.0 (* (* (* k k) (* k (/ t l))) t_1))
     (/
      2.0
      (*
       (* t_1 (* (* (* (fma -0.3333333333333333 (* k k) 1.0) t) k) k))
       (/ k l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / cos(k)) / l;
	double tmp;
	if (t <= 1.5e-15) {
		tmp = 2.0 / (((k * k) * (k * (t / l))) * t_1);
	} else {
		tmp = 2.0 / ((t_1 * (((fma(-0.3333333333333333, (k * k), 1.0) * t) * k) * k)) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / cos(k)) / l)
	tmp = 0.0
	if (t <= 1.5e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(k * Float64(t / l))) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * t) * k) * k)) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 1.5e-15], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[(N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e-15

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 72.9%

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

   10. Applied rewrites 74.4%

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

   if 1.5e-15 < t

   1. Initial program 25.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 87.7%

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

   7. Applied rewrites 95.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 81.3%

       \[\leadsto \frac{2}{\left(\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot t\right) \cdot k\right) \cdot k\right)\right) \cdot \frac{k}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.2% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ (/ k (cos k)) l) (* (* k k) t)) (/ k l))))

C

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

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 / ((((k / cos(k)) / l) * ((k * k) * t)) * (k / l))
end function

Java

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

Python

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

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64(k / cos(k)) / l) * Float64(Float64(k * k) * t)) * Float64(k / l)))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 90.2%

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

7. Applied rewrites 96.1%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 76.3%

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

Alternative 10: 73.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot 2}{k} \cdot \frac{\frac{\ell}{k \cdot k}}{k \cdot t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* l 2.0) k) (/ (/ l (* k k)) (* k t))))

C

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

Fortran

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) * ((l / (k * k)) / (k * t))
end function

Java

public static double code(double t, double l, double k) {
	return ((l * 2.0) / k) * ((l / (k * k)) / (k * t));
}

Python

def code(t, l, k):
	return ((l * 2.0) / k) * ((l / (k * k)) / (k * t))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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 69.2

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

5. Applied rewrites 69.2%

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

7. Applied rewrites 71.4%

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

9. Applied rewrites 75.7%

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

11. Applied rewrites 76.0%

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

Alternative 11: 72.4% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ (/ l (* k k)) (* (* k t) k))))

C

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

Fortran

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) * ((l / (k * k)) / ((k * t) * k))
end function

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * ((l / (k * k)) / ((k * t) * k));
}

Python

def code(t, l, k):
	return (l * 2.0) * ((l / (k * k)) / ((k * t) * k))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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 69.2

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

5. Applied rewrites 69.2%

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

7. Applied rewrites 71.4%

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

9. Applied rewrites 75.7%

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

11. Applied rewrites 75.7%

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

Alternative 12: 72.4% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ (/ l (* k k)) (* (* k k) t))))

C

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

Fortran

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) * ((l / (k * k)) / ((k * k) * t))
end function

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * ((l / (k * k)) / ((k * k) * t));
}

Python

def code(t, l, k):
	return (l * 2.0) * ((l / (k * k)) / ((k * k) * t))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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 69.2

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

5. Applied rewrites 69.2%

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

7. Applied rewrites 71.4%

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

9. Applied rewrites 75.7%

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

Alternative 13: 71.8% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ (/ l k) (* (* (* k k) t) k))))

C

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

Fortran

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) * ((l / k) / (((k * k) * t) * k))
end function

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * ((l / k) / (((k * k) * t) * k));
}

Python

def code(t, l, k):
	return (l * 2.0) * ((l / k) / (((k * k) * t) * k))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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 69.2

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

5. Applied rewrites 69.2%

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

7. Applied rewrites 71.4%

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

9. Applied rewrites 75.7%

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

11. Applied rewrites 74.7%

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

Alternative 14: 70.5% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l 2.0) (/ l (* (* (* k k) t) (* k k)))))

C

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

Fortran

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) * (l / (((k * k) * t) * (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * 2.0) * (l / (((k * k) * t) * (k * k)));
}

Python

def code(t, l, k):
	return (l * 2.0) * (l / (((k * k) * t) * (k * k)))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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 69.2

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

5. Applied rewrites 69.2%

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

7. Applied rewrites 71.4%

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

9. Applied rewrites 75.7%

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

11. Applied rewrites 73.7%

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

12. Final simplification 73.7%

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

Reproduce

herbie shell --seed 2024322 
(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))))