Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 98.1%

Time: 16.0s

Alternatives: 13

Speedup: 11.6×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 28.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. pow-flip N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. cube-unmult N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*r* N/A

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

6. Applied rewrites 93.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 95.2

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

8. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. lower-/.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 98.5

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

10. Applied rewrites 98.5%

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

Alternative 2: 93.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 28.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. pow-flip N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. cube-unmult N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*r* N/A

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

6. Applied rewrites 93.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 95.2

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

8. Applied rewrites 95.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 96.0

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

10. Applied rewrites 96.0%

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

11. Final simplification 96.0%

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

Alternative 3: 84.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k}}{t \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6e-21)
   (/ (/ (+ l l) (* k k)) (* t (/ (* k k) l)))
   (* (/ (+ l l) k) (/ l (* (* t k) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-21) {
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l));
	} else {
		tmp = ((l + l) / k) * (l / ((t * k) * (sin(k) * tan(k))));
	}
	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 (k <= 6d-21) then
        tmp = ((l + l) / (k * k)) / (t * ((k * k) / l))
    else
        tmp = ((l + l) / k) * (l / ((t * k) * (sin(k) * tan(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-21) {
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l));
	} else {
		tmp = ((l + l) / k) * (l / ((t * k) * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6e-21:
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l))
	else:
		tmp = ((l + l) / k) * (l / ((t * k) * (math.sin(k) * math.tan(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6e-21)
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) / Float64(t * Float64(Float64(k * k) / l)));
	else
		tmp = Float64(Float64(Float64(l + l) / k) * Float64(l / Float64(Float64(t * k) * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6e-21)
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l));
	else
		tmp = ((l + l) / k) * (l / ((t * k) * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6e-21], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[(t * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999982e-21

   1. Initial program 36.0%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 62.9

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 69.8%

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

   9. Applied rewrites 74.4%

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

   11. Applied rewrites 81.0%

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

   if 5.99999999999999982e-21 < k

   1. Initial program 30.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 30.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-flip N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

   6. Applied rewrites 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 97.8

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

   8. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. count-2 N/A

          \[\leadsto \frac{\color{blue}{\ell + \ell}}{k} \cdot \frac{\frac{\ell}{t \cdot k}}{\sin k \cdot \tan k} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\ell + \ell}}{k} \cdot \frac{\frac{\ell}{t \cdot k}}{\sin k \cdot \tan k} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\color{blue}{\frac{\ell}{t \cdot k}}}{\sin k \cdot \tan k} \]

      16. associate-/l/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 97.8

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

   10. Applied rewrites 97.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k}}{t \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 18500:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k}}{t \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{t \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 18500.0)
   (/ (/ (+ l l) (* k k)) (* t (/ (* k k) l)))
   (/ (/ (/ (* 2.0 (* l l)) (- 0.5 (* 0.5 (cos (+ k k))))) (* t k)) k)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 18500.0) {
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l));
	} else {
		tmp = (((2.0 * (l * l)) / (0.5 - (0.5 * cos((k + k))))) / (t * k)) / k;
	}
	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 (k <= 18500.0d0) then
        tmp = ((l + l) / (k * k)) / (t * ((k * k) / l))
    else
        tmp = (((2.0d0 * (l * l)) / (0.5d0 - (0.5d0 * cos((k + k))))) / (t * k)) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 18500.0) {
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l));
	} else {
		tmp = (((2.0 * (l * l)) / (0.5 - (0.5 * Math.cos((k + k))))) / (t * k)) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 18500.0:
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l))
	else:
		tmp = (((2.0 * (l * l)) / (0.5 - (0.5 * math.cos((k + k))))) / (t * k)) / k
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 18500.0)
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) / Float64(t * Float64(Float64(k * k) / l)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(l * l)) / Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))) / Float64(t * k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 18500.0)
		tmp = ((l + l) / (k * k)) / (t * ((k * k) / l));
	else
		tmp = (((2.0 * (l * l)) / (0.5 - (0.5 * cos((k + k))))) / (t * k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 18500.0], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 18500

   1. Initial program 35.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 80.6%

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

   if 18500 < k

   1. Initial program 31.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 57.9%

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

   10. Applied rewrites 58.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 18500:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k}}{t \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-169}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell + \ell\right)}{\left(t \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e-169)
   (* (/ (+ l l) (* t k)) (/ l (* k (* k k))))
   (/ (* (/ l k) (+ l l)) (* (* t k) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-169) {
		tmp = ((l + l) / (t * k)) * (l / (k * (k * k)));
	} else {
		tmp = ((l / k) * (l + l)) / ((t * k) * (k * k));
	}
	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-169) then
        tmp = ((l + l) / (t * k)) * (l / (k * (k * k)))
    else
        tmp = ((l / k) * (l + l)) / ((t * k) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e-169) {
		tmp = ((l + l) / (t * k)) * (l / (k * (k * k)));
	} else {
		tmp = ((l / k) * (l + l)) / ((t * k) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e-169:
		tmp = ((l + l) / (t * k)) * (l / (k * (k * k)))
	else:
		tmp = ((l / k) * (l + l)) / ((t * k) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e-169)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * k)) * Float64(l / Float64(k * Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l + l)) / Float64(Float64(t * k) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e-169)
		tmp = ((l + l) / (t * k)) * (l / (k * (k * k)));
	else
		tmp = ((l / k) * (l + l)) / ((t * k) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e-169], N[(N[(N[(l + l), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[(t * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.00000000000000004e-169

   1. Initial program 27.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 80.9%

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

   11. Applied rewrites 80.9%

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

   if 2.00000000000000004e-169 < (*.f64 l l)

   1. Initial program 37.3%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 58.5%

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

   9. Applied rewrites 59.1%

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

   11. Applied rewrites 62.9%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-169}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell + \ell\right)}{\left(t \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

9. Applied rewrites 66.6%

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

11. Applied rewrites 71.6%

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

12. Final simplification 71.6%

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

Alternative 7: 74.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

9. Applied rewrites 66.6%

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

11. Applied rewrites 71.6%

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

Alternative 8: 73.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 70.3%

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

8. Final simplification 70.3%

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

Alternative 9: 72.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

9. Applied rewrites 66.6%

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

11. Applied rewrites 69.9%

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

12. Final simplification 69.9%

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

Alternative 10: 70.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

8. Taylor expanded in l around 0

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

10. Applied rewrites 63.7%

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

12. Applied rewrites 67.3%

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

13. Final simplification 67.3%

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

Alternative 11: 70.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

8. Taylor expanded in l around 0

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

10. Applied rewrites 63.7%

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

12. Applied rewrites 67.3%

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

13. Final simplification 67.3%

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

Alternative 12: 69.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

8. Taylor expanded in l around 0

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

10. Applied rewrites 63.7%

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

12. Applied rewrites 65.1%

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

Alternative 13: 68.5% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.0%

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

3. Taylor expanded in 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 63.7%

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

8. Taylor expanded in l around 0

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

10. Applied rewrites 63.7%

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

11. Taylor expanded in t around 0

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

13. Applied rewrites 63.7%

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

Reproduce

herbie shell --seed 2024233 
(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))))