Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 92.4%

Time: 17.2s

Alternatives: 21

Speedup: 18.5×

• 39.6% 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: 35.6% 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: 92.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.5%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. count-2 N/A

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

   12. lower-+.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-pow.f64 N/A

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

   21. lower-sin.f64 74.5

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

5. Simplified 74.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

7. Applied egg-rr 81.7%

   \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
8. Step-by-step derivation

   1. count-2 N/A

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

   2. sqr-sin-a N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-sin.f64 93.4

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

9. Applied egg-rr 93.4%

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

Alternative 2: 92.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. count-2 N/A

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

   12. lower-+.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-pow.f64 N/A

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

   21. lower-sin.f64 76.3

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

5. Simplified 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

7. Applied egg-rr 83.6%

   \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
8. Step-by-step derivation

   1. count-2 N/A

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

   2. sqr-sin-a N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-sin.f64 92.4

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

9. Applied egg-rr 92.4%

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

    1. lift-cos.f64 N/A

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

    2. *-commutative N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 92.4

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

11. Applied egg-rr 92.4%

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

12. Final simplification 92.4%

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

Reproduce

herbie shell --seed 2024218 
(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))))