Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.7% → 97.0%

Time: 12.2s

Alternatives: 19

Speedup: 11.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 92.5%

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

7. Applied rewrites 97.6%

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

9. Applied rewrites 98.7%

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

Alternative 2: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \leq -1 \cdot 10^{-198}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot \mathsf{fma}\left(-0.3333333333333333 \cdot k, k, 2\right)}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (- (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
        -1e-198)
     (/ (* (* l (/ l t)) (fma (* -0.3333333333333333 k) k 2.0)) (pow k 4.0))
     (/ 2.0 (* (* t_1 t_1) t)))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0))) <= -1e-198) {
		tmp = ((l * (l / t)) * fma((-0.3333333333333333 * k), k, 2.0)) / pow(k, 4.0);
	} else {
		tmp = 2.0 / ((t_1 * t_1) * t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (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))) <= -1e-198)
		tmp = Float64(Float64(Float64(l * Float64(l / t)) * fma(Float64(-0.3333333333333333 * k), k, 2.0)) / (k ^ 4.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -1e-198], N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.3333333333333333 * k), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -9.9999999999999991e-199

   1. Initial program 76.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 99.7%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 66.5%

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

   if -9.9999999999999991e-199 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 28.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 74.1%

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

Alternative 3: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot {\sin k}^{2}\right) \cdot k}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{0.5}{\frac{\ell}{k}} - \frac{\cos \left(k \cdot 2\right) \cdot 0.5}{\frac{\ell}{k}}\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e+149)
   (/ 2.0 (/ (* (* (* t (/ k l)) (pow (sin k) 2.0)) k) (* (cos k) l)))
   (/
    2.0
    (*
     (* t (- (/ 0.5 (/ l k)) (/ (* (cos (* k 2.0)) 0.5) (/ l k))))
     (/ (/ k (cos k)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+149) {
		tmp = 2.0 / ((((t * (k / l)) * pow(sin(k), 2.0)) * k) / (cos(k) * l));
	} else {
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / cos(k)) / l));
	}
	return tmp;
}

Fortran

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 ((l * l) <= 2d+149) then
        tmp = 2.0d0 / ((((t * (k / l)) * (sin(k) ** 2.0d0)) * k) / (cos(k) * l))
    else
        tmp = 2.0d0 / ((t * ((0.5d0 / (l / k)) - ((cos((k * 2.0d0)) * 0.5d0) / (l / k)))) * ((k / cos(k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+149) {
		tmp = 2.0 / ((((t * (k / l)) * Math.pow(Math.sin(k), 2.0)) * k) / (Math.cos(k) * l));
	} else {
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((Math.cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e+149:
		tmp = 2.0 / ((((t * (k / l)) * math.pow(math.sin(k), 2.0)) * k) / (math.cos(k) * l))
	else:
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((math.cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+149)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(k / l)) * (sin(k) ^ 2.0)) * k) / Float64(cos(k) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(0.5 / Float64(l / k)) - Float64(Float64(cos(Float64(k * 2.0)) * 0.5) / Float64(l / k)))) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+149)
		tmp = 2.0 / ((((t * (k / l)) * (sin(k) ^ 2.0)) * k) / (cos(k) * l));
	else
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+149], N[(2.0 / N[(N[(N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(0.5 / N[(l / k), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000001e149

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 92.8%

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

   7. Applied rewrites 96.8%

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

   9. Applied rewrites 98.6%

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

   11. Applied rewrites 96.1%

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

   if 2.0000000000000001e149 < (*.f64 l l)

   1. Initial program 32.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 92.1%

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

   7. Applied rewrites 98.8%

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

   9. Applied rewrites 75.4%

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

Alternative 4: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-40}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)\right) \cdot t}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{0.5}{\frac{\ell}{k}} - \frac{\cos \left(k \cdot 2\right) \cdot 0.5}{\frac{\ell}{k}}\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-40)
   (/ 2.0 (/ (* (* k (* (/ k l) (pow (sin k) 2.0))) t) (* (cos k) l)))
   (/
    2.0
    (*
     (* t (- (/ 0.5 (/ l k)) (/ (* (cos (* k 2.0)) 0.5) (/ l k))))
     (/ (/ k (cos k)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-40) {
		tmp = 2.0 / (((k * ((k / l) * pow(sin(k), 2.0))) * t) / (cos(k) * l));
	} else {
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / cos(k)) / l));
	}
	return tmp;
}

Fortran

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 ((l * l) <= 1d-40) then
        tmp = 2.0d0 / (((k * ((k / l) * (sin(k) ** 2.0d0))) * t) / (cos(k) * l))
    else
        tmp = 2.0d0 / ((t * ((0.5d0 / (l / k)) - ((cos((k * 2.0d0)) * 0.5d0) / (l / k)))) * ((k / cos(k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-40) {
		tmp = 2.0 / (((k * ((k / l) * Math.pow(Math.sin(k), 2.0))) * t) / (Math.cos(k) * l));
	} else {
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((Math.cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-40:
		tmp = 2.0 / (((k * ((k / l) * math.pow(math.sin(k), 2.0))) * t) / (math.cos(k) * l))
	else:
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((math.cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-40)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(Float64(k / l) * (sin(k) ^ 2.0))) * t) / Float64(cos(k) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(0.5 / Float64(l / k)) - Float64(Float64(cos(Float64(k * 2.0)) * 0.5) / Float64(l / k)))) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-40)
		tmp = 2.0 / (((k * ((k / l) * (sin(k) ^ 2.0))) * t) / (cos(k) * l));
	else
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-40], N[(2.0 / N[(N[(N[(k * N[(N[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(0.5 / N[(l / k), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999993e-41

   1. Initial program 30.6%

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

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

   9. Applied rewrites 94.9%

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

   if 9.9999999999999993e-41 < (*.f64 l l)

   1. Initial program 36.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 94.3%

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 71.9%

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

Alternative 5: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 / ((t * (pow(sin(k), 2.0) * (k / l))) * ((k / cos(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 / ((t * ((sin(k) ** 2.0d0) * (k / l))) * ((k / cos(k)) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 92.5%

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

7. Applied rewrites 97.6%

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

Alternative 6: 78.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{0.5}{\frac{\ell}{k}} - \frac{\cos \left(k \cdot 2\right) \cdot 0.5}{\frac{\ell}{k}}\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= (* l l) 5e-134)
     (/ 2.0 (* (* t_1 t_1) t))
     (/
      2.0
      (*
       (* t (- (/ 0.5 (/ l k)) (/ (* (cos (* k 2.0)) 0.5) (/ l k))))
       (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((l * l) <= 5e-134) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / cos(k)) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k / l)
    if ((l * l) <= 5d-134) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / ((t * ((0.5d0 / (l / k)) - ((cos((k * 2.0d0)) * 0.5d0) / (l / k)))) * ((k / cos(k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((l * l) <= 5e-134) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((Math.cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k / l)
	tmp = 0
	if (l * l) <= 5e-134:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((math.cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (Float64(l * l) <= 5e-134)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(0.5 / Float64(l / k)) - Float64(Float64(cos(Float64(k * 2.0)) * 0.5) / Float64(l / k)))) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 0.0;
	if ((l * l) <= 5e-134)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / ((t * ((0.5 / (l / k)) - ((cos((k * 2.0)) * 0.5) / (l / k)))) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-134], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(0.5 / N[(l / k), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000003e-134

   1. Initial program 27.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 85.0

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

   5. Applied rewrites 85.0%

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

   7. Applied rewrites 92.4%

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

   if 5.0000000000000003e-134 < (*.f64 l l)

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 94.8%

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 71.9%

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

Alternative 7: 83.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 1.42 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k}{\ell} \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= k 1.42e-6)
     (/ 2.0 (* (* t_1 t_1) t))
     (/
      2.0
      (*
       (/ (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t) k) l)
       (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 1.42e-6) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (((((0.5 - (0.5 * cos((k + k)))) * t) * k) / l) * ((k / cos(k)) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k / l)
    if (k <= 1.42d-6) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / (((((0.5d0 - (0.5d0 * cos((k + k)))) * t) * k) / l) * ((k / cos(k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 1.42e-6) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (((((0.5 - (0.5 * Math.cos((k + k)))) * t) * k) / l) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k / l)
	tmp = 0
	if k <= 1.42e-6:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / (((((0.5 - (0.5 * math.cos((k + k)))) * t) * k) / l) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (k <= 1.42e-6)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t) * k) / l) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 0.0;
	if (k <= 1.42e-6)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / (((((0.5 - (0.5 * cos((k + k)))) * t) * k) / l) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.42e-6], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $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] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.42e-6

   1. Initial program 34.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 81.9%

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

   if 1.42e-6 < k

   1. Initial program 31.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 93.1%

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

   7. Applied rewrites 92.8%

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

Alternative 8: 83.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 1.42 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\cos \left(2 \cdot k\right), -0.5, 0.5\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= k 1.42e-6)
     (/ 2.0 (* (* t_1 t_1) t))
     (/
      2.0
      (*
       (* (fma (cos (* 2.0 k)) -0.5 0.5) (* (/ t l) k))
       (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 1.42e-6) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((fma(cos((2.0 * k)), -0.5, 0.5) * ((t / l) * k)) * ((k / cos(k)) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (k <= 1.42e-6)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(cos(Float64(2.0 * k)), -0.5, 0.5) * Float64(Float64(t / l) * k)) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.42e-6], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.42e-6

   1. Initial program 34.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 81.9%

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

   if 1.42e-6 < k

   1. Initial program 31.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 93.1%

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 90.6%

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

Alternative 9: 77.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\cos \left(2 \cdot k\right), -0.5, 0.5\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= k 2.65e-5)
     (/ 2.0 (* (* t_1 t_1) t))
     (/
      2.0
      (*
       (fma (cos (* 2.0 k)) -0.5 0.5)
       (/ (* (* k k) t) (* (* (cos k) l) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 2.65e-5) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (fma(cos((2.0 * k)), -0.5, 0.5) * (((k * k) * t) / ((cos(k) * l) * l)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (k <= 2.65e-5)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(fma(cos(Float64(2.0 * k)), -0.5, 0.5) * Float64(Float64(Float64(k * k) * t) / Float64(Float64(cos(k) * l) * l))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.65e-5], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.65e-5

   1. Initial program 34.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 81.9%

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

   if 2.65e-5 < k

   1. Initial program 31.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 93.1%

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 67.8%

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

Alternative 10: 74.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 8 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{{k}^{3}}{\ell}\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= k 8e-70)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (* (* t (/ (pow k 3.0) l)) (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 8e-70) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((t * (pow(k, 3.0) / l)) * ((k / cos(k)) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k / l)
    if (k <= 8d-70) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / ((t * ((k ** 3.0d0) / l)) * ((k / cos(k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 8e-70) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / ((t * (Math.pow(k, 3.0) / l)) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k / l)
	tmp = 0
	if k <= 8e-70:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / ((t * (math.pow(k, 3.0) / l)) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (k <= 8e-70)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64((k ^ 3.0) / l)) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 0.0;
	if (k <= 8e-70)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / ((t * ((k ^ 3.0) / l)) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 8e-70], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Power[k, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.99999999999999997e-70

   1. Initial program 36.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 81.2%

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

   if 7.99999999999999997e-70 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 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. Step-by-step derivation

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 60.8%

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

Alternative 11: 74.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq 1.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}{\ell} \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (<= k 1.6e-36)
     (/ 2.0 (* (* t_1 t_1) t))
     (/ 2.0 (* (/ (* (* (* k k) t) k) l) (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 1.6e-36) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (((((k * k) * t) * k) / l) * ((k / cos(k)) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k / l)
    if (k <= 1.6d-36) then
        tmp = 2.0d0 / ((t_1 * t_1) * t)
    else
        tmp = 2.0d0 / (((((k * k) * t) * k) / l) * ((k / cos(k)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if (k <= 1.6e-36) {
		tmp = 2.0 / ((t_1 * t_1) * t);
	} else {
		tmp = 2.0 / (((((k * k) * t) * k) / l) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k / l)
	tmp = 0
	if k <= 1.6e-36:
		tmp = 2.0 / ((t_1 * t_1) * t)
	else:
		tmp = 2.0 / (((((k * k) * t) * k) / l) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if (k <= 1.6e-36)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t_1) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * k) / l) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 0.0;
	if (k <= 1.6e-36)
		tmp = 2.0 / ((t_1 * t_1) * t);
	else
		tmp = 2.0 / (((((k * k) * t) * k) / l) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.6e-36], N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.60000000000000011e-36

   1. Initial program 34.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 81.6%

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

   if 1.60000000000000011e-36 < k

   1. Initial program 31.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 56.4%

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

Alternative 12: 71.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot {\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.8e-123)
   (* (* l (/ 2.0 t)) (/ (/ l (* k k)) (* k k)))
   (* (* l 2.0) (pow (* (* k (/ k l)) (* (* k k) t)) -1.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.8e-123) {
		tmp = (l * (2.0 / t)) * ((l / (k * k)) / (k * k));
	} else {
		tmp = (l * 2.0) * pow(((k * (k / l)) * ((k * k) * t)), -1.0);
	}
	return tmp;
}

Fortran

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 <= 5.8d-123) then
        tmp = (l * (2.0d0 / t)) * ((l / (k * k)) / (k * k))
    else
        tmp = (l * 2.0d0) * (((k * (k / l)) * ((k * k) * t)) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.8e-123) {
		tmp = (l * (2.0 / t)) * ((l / (k * k)) / (k * k));
	} else {
		tmp = (l * 2.0) * Math.pow(((k * (k / l)) * ((k * k) * t)), -1.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.8e-123:
		tmp = (l * (2.0 / t)) * ((l / (k * k)) / (k * k))
	else:
		tmp = (l * 2.0) * math.pow(((k * (k / l)) * ((k * k) * t)), -1.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.8e-123)
		tmp = Float64(Float64(l * Float64(2.0 / t)) * Float64(Float64(l / Float64(k * k)) / Float64(k * k)));
	else
		tmp = Float64(Float64(l * 2.0) * (Float64(Float64(k * Float64(k / l)) * Float64(Float64(k * k) * t)) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.8e-123)
		tmp = (l * (2.0 / t)) * ((l / (k * k)) / (k * k));
	else
		tmp = (l * 2.0) * (((k * (k / l)) * ((k * k) * t)) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.8e-123], N[(N[(l * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * N[Power[N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.80000000000000007e-123

   1. Initial program 26.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 67.8

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 68.5%

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

   9. Applied rewrites 68.5%

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

   if 5.80000000000000007e-123 < t

   1. Initial program 46.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

   7. Applied rewrites 69.9%

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

   9. Applied rewrites 76.2%

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

   11. Applied rewrites 76.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{t}\right) \cdot \frac{\frac{\ell}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot {\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \frac{2}{\left(t\_1 \cdot t\_1\right) \cdot t} \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l)))) (/ 2.0 (* (* t_1 t_1) t))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	return 2.0 / ((t_1 * t_1) * t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    t_1 = k * (k / l)
    code = 2.0d0 / ((t_1 * t_1) * t)
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	return 2.0 / ((t_1 * t_1) * t);
}

Python

def code(t, l, k):
	t_1 = k * (k / l)
	return 2.0 / ((t_1 * t_1) * t)

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	return Float64(2.0 / Float64(Float64(t_1 * t_1) * t))
end

MATLAB

function tmp = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 2.0 / ((t_1 * t_1) * t);
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(2.0 / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 69.5

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

5. Applied rewrites 69.5%

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

7. Applied rewrites 72.5%

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

Alternative 14: 74.3% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in 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 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 70.5%

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

11. Applied rewrites 72.2%

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

Alternative 15: 73.6% 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 33.8%

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

3. Taylor expanded in 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 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 70.5%

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

11. Applied rewrites 71.4%

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

Alternative 16: 72.2% accurate, 9.6× speedup

Math

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

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(Float64(l / 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[(N[(l / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in 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 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 70.5%

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

11. Applied rewrites 70.5%

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

Alternative 17: 72.3% 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 33.8%

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

3. Taylor expanded in 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 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 70.5%

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

11. Applied rewrites 70.5%

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

Alternative 18: 72.2% 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 33.8%

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

3. Taylor expanded in 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 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 70.5%

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

Alternative 19: 70.3% 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 33.8%

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

3. Taylor expanded in 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 68.2

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

5. Applied rewrites 68.2%

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

7. Applied rewrites 69.0%

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

9. Applied rewrites 70.5%

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

11. Applied rewrites 68.8%

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

12. Final simplification 68.8%

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

Reproduce

herbie shell --seed 2024323 
(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))))