Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 98.1%

Time: 15.9s

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 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. 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 28.9%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{t \cdot t} \cdot \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
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 92.5%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
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 93.4

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

8. Applied rewrites 93.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{\ell}{t \cdot k}}{\frac{k}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
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 93.9

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

10. Applied rewrites 93.9%

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

11. Final simplification 93.9%

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