Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 92.5%

Time: 30.0s

Alternatives: 6

Speedup: 24.8×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.9% 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: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 1.4e-105)
   (* 2.0 (/ (/ (pow (/ l k) 2.0) k) (* k t)))
   (*
    2.0
    (* (* (/ l k) (/ l k)) (* (cos k) (/ 1.0 (* t (pow (sin k) 2.0))))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.4e-105) {
		tmp = 2.0 * ((pow((l / k), 2.0) / k) / (k * t));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * (1.0 / (t * pow(sin(k), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 <= 1.4d-105) then
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / k) / (k * t))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) * (1.0d0 / (t * (sin(k) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.4e-105) {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / k) / (k * t));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) * (1.0 / (t * Math.pow(Math.sin(k), 2.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if l <= 1.4e-105:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / k) / (k * t))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) * (1.0 / (t * math.pow(math.sin(k), 2.0)))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 1.4e-105)
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / k) / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) * Float64(1.0 / Float64(t * (sin(k) ^ 2.0))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.4e-105)
		tmp = 2.0 * ((((l / k) ^ 2.0) / k) / (k * t));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * (1.0 / (t * (sin(k) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 1.4e-105], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(1.0 / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e-105

   1. Initial program 28.6%

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

      1. associate-/r* 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-/r* 30.9%

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

      4. associate-*r/ 32.6%

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

      5. associate-/l* 30.9%

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

      6. +-commutative 30.9%

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

      7. unpow2 30.9%

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

      8. sqr-neg 30.9%

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

      9. distribute-frac-neg 30.9%

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

      10. distribute-frac-neg 30.9%

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

      11. unpow2 30.9%

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

      12. associate--l+ 42.3%

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

      13. metadata-eval 42.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 42.3%

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

      15. unpow2 42.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      16. distribute-frac-neg 42.3%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in k around inf 69.7%

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

      1. *-commutative 69.7%

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

      2. times-frac 70.6%

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

      3. unpow2 70.6%

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

      4. unpow2 70.6%

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

      5. times-frac 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in k around 0 72.4%

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

      1. unpow2 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in k around 0 72.4%

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

       1. unpow2 72.4%

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

       2. associate-*r* 75.3%

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

   12. Simplified 75.3%

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

       1. div-inv 75.9%

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

       2. associate-/r* 76.9%

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

       3. pow2 76.9%

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

   14. Applied egg-rr 76.9%

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

   if 1.4e-105 < l

   1. Initial program 39.1%

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

      1. associate-/r* 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-/r* 41.5%

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

      4. associate-*r/ 41.5%

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

      5. associate-/l* 41.5%

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

      6. +-commutative 41.5%

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

      7. unpow2 41.5%

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

      8. sqr-neg 41.5%

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

      9. distribute-frac-neg 41.5%

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

      10. distribute-frac-neg 41.5%

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

      11. unpow2 41.5%

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

      12. associate--l+ 53.1%

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

      13. metadata-eval 53.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 53.1%

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

      15. unpow2 53.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      16. distribute-frac-neg 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in k around inf 76.9%

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

      1. *-commutative 76.9%

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

      2. times-frac 77.0%

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

      3. unpow2 77.0%

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

      4. unpow2 77.0%

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

      5. times-frac 94.8%

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

   6. Simplified 94.8%

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

      1. div-inv 94.9%

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

   8. Applied egg-rr 94.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right)\\ \end{array} \]

Alternative 2: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-104}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 1.05e-104)
   (* 2.0 (/ (/ (pow (/ l k) 2.0) k) (* k t)))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.05e-104) {
		tmp = 2.0 * ((pow((l / k), 2.0) / k) / (k * t));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
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 <= 1.05d-104) then
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / k) / (k * t))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.05e-104) {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / k) / (k * t));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if l <= 1.05e-104:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / k) / (k * t))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 1.05e-104)
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / k) / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.05e-104)
		tmp = 2.0 * ((((l / k) ^ 2.0) / k) / (k * t));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 1.05e-104], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.04999999999999999e-104

   1. Initial program 28.6%

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

      1. associate-/r* 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-/r* 30.9%

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

      4. associate-*r/ 32.6%

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

      5. associate-/l* 30.9%

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

      6. +-commutative 30.9%

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

      7. unpow2 30.9%

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

      8. sqr-neg 30.9%

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

      9. distribute-frac-neg 30.9%

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

      10. distribute-frac-neg 30.9%

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

      11. unpow2 30.9%

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

      12. associate--l+ 42.3%

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

      13. metadata-eval 42.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 42.3%

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

      15. unpow2 42.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      16. distribute-frac-neg 42.3%

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

   3. Simplified 42.3%

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

   4. Taylor expanded in k around inf 69.7%

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

      1. *-commutative 69.7%

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

      2. times-frac 70.6%

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

      3. unpow2 70.6%

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

      4. unpow2 70.6%

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

      5. times-frac 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in k around 0 72.4%

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

      1. unpow2 72.4%

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

   9. Simplified 72.4%

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

   10. Taylor expanded in k around 0 72.4%

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

       1. unpow2 72.4%

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

       2. associate-*r* 75.3%

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

   12. Simplified 75.3%

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

       1. div-inv 75.9%

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

       2. associate-/r* 76.9%

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

       3. pow2 76.9%

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

   14. Applied egg-rr 76.9%

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

   if 1.04999999999999999e-104 < l

   1. Initial program 39.1%

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

      1. associate-/r* 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-/r* 41.5%

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

      4. associate-*r/ 41.5%

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

      5. associate-/l* 41.5%

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

      6. +-commutative 41.5%

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

      7. unpow2 41.5%

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

      8. sqr-neg 41.5%

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

      9. distribute-frac-neg 41.5%

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

      10. distribute-frac-neg 41.5%

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

      11. unpow2 41.5%

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

      12. associate--l+ 53.1%

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

      13. metadata-eval 53.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 53.1%

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

      15. unpow2 53.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      16. distribute-frac-neg 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in k around inf 76.9%

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

      1. *-commutative 76.9%

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

      2. times-frac 77.0%

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

      3. unpow2 77.0%

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

      4. unpow2 77.0%

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

      5. times-frac 94.8%

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

   6. Simplified 94.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-104}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 71.0% accurate, 3.8× speedup

Math

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

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (/ (pow (/ l k) 2.0) k) (* k t))))

C

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

Fortran

NOTE: l should be positive before calling this function
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 / k) ** 2.0d0) / k) / (k * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 34.2%

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

   4. associate-*r/ 35.4%

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

   5. associate-/l* 34.2%

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

   6. +-commutative 34.2%

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

   7. unpow2 34.2%

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

   8. sqr-neg 34.2%

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

   9. distribute-frac-neg 34.2%

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

   10. distribute-frac-neg 34.2%

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

   11. unpow2 34.2%

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

   12. associate--l+ 45.7%

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

   13. metadata-eval 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.7%

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

   15. unpow2 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in k around inf 72.0%

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

   1. *-commutative 72.0%

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

   2. times-frac 72.6%

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

   3. unpow2 72.6%

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

   4. unpow2 72.6%

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

   5. times-frac 91.4%

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

6. Simplified 91.4%

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

7. Taylor expanded in k around 0 70.3%

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

   1. unpow2 70.3%

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

9. Simplified 70.3%

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

10. Taylor expanded in k around 0 70.3%

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

    1. unpow2 70.3%

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

    2. associate-*r* 72.3%

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

12. Simplified 72.3%

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

    1. div-inv 72.7%

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

    2. associate-/r* 73.4%

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

    3. pow2 73.4%

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

14. Applied egg-rr 73.4%

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

15. Final simplification 73.4%

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

Alternative 4: 70.3% accurate, 24.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (* (/ l k) (/ l k)) (/ 1.0 (* k (* k t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))));
}

Fortran

NOTE: l should be positive before calling this function
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 / k) * (l / k)) * (1.0d0 / (k * (k * t))))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))));
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(1.0 / Float64(k * Float64(k * t)))))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 34.2%

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

   4. associate-*r/ 35.4%

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

   5. associate-/l* 34.2%

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

   6. +-commutative 34.2%

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

   7. unpow2 34.2%

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

   8. sqr-neg 34.2%

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

   9. distribute-frac-neg 34.2%

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

   10. distribute-frac-neg 34.2%

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

   11. unpow2 34.2%

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

   12. associate--l+ 45.7%

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

   13. metadata-eval 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.7%

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

   15. unpow2 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in k around inf 72.0%

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

   1. *-commutative 72.0%

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

   2. times-frac 72.6%

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

   3. unpow2 72.6%

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

   4. unpow2 72.6%

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

   5. times-frac 91.4%

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

6. Simplified 91.4%

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

7. Taylor expanded in k around 0 70.3%

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

   1. unpow2 70.3%

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

9. Simplified 70.3%

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

10. Taylor expanded in k around 0 70.3%

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

    1. unpow2 70.3%

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

    2. associate-*r* 72.3%

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

12. Simplified 72.3%

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

13. Final simplification 72.3%

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

Alternative 5: 70.3% accurate, 24.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k}}{k \cdot t}\right) \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (* (/ l k) (/ l k)) (/ (/ 1.0 k) (* k t)))))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * ((1.0 / k) / (k * t)));
}

Fortran

NOTE: l should be positive before calling this function
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 / k) * (l / k)) * ((1.0d0 / k) / (k * t)))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * ((1.0 / k) / (k * t)));
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 * (((l / k) * (l / k)) * ((1.0 / k) / (k * t)))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / k) / Float64(k * t))))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / k) / (k * t)));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 34.2%

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

   4. associate-*r/ 35.4%

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

   5. associate-/l* 34.2%

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

   6. +-commutative 34.2%

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

   7. unpow2 34.2%

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

   8. sqr-neg 34.2%

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

   9. distribute-frac-neg 34.2%

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

   10. distribute-frac-neg 34.2%

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

   11. unpow2 34.2%

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

   12. associate--l+ 45.7%

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

   13. metadata-eval 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.7%

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

   15. unpow2 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in k around inf 72.0%

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

   1. *-commutative 72.0%

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

   2. times-frac 72.6%

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

   3. unpow2 72.6%

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

   4. unpow2 72.6%

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

   5. times-frac 91.4%

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

6. Simplified 91.4%

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

7. Taylor expanded in k around 0 70.3%

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

   1. unpow2 70.3%

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

9. Simplified 70.3%

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

10. Taylor expanded in k around 0 70.3%

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

    1. unpow2 70.3%

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

    2. associate-*r* 72.3%

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

12. Simplified 72.3%

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

13. Taylor expanded in k around 0 70.3%

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

    1. unpow2 70.3%

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

    2. associate-*r* 72.3%

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

    3. associate-/r* 72.3%

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

15. Simplified 72.3%

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

16. Final simplification 72.3%

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

Alternative 6: 33.2% accurate, 38.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ (* l l) (* k (* k t))) -0.3333333333333333))

C

l = abs(l);
double code(double t, double l, double k) {
	return ((l * l) / (k * (k * t))) * -0.3333333333333333;
}

Fortran

NOTE: l should be positive before calling this function
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))) * (-0.3333333333333333d0)
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return ((l * l) / (k * (k * t))) * -0.3333333333333333;
}

Python

l = abs(l)
def code(t, l, k):
	return ((l * l) / (k * (k * t))) * -0.3333333333333333

Julia

l = abs(l)
function code(t, l, k)
	return Float64(Float64(Float64(l * l) / Float64(k * Float64(k * t))) * -0.3333333333333333)
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = ((l * l) / (k * (k * t))) * -0.3333333333333333;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 34.2%

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

   4. associate-*r/ 35.4%

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

   5. associate-/l* 34.2%

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

   6. +-commutative 34.2%

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

   7. unpow2 34.2%

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

   8. sqr-neg 34.2%

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

   9. distribute-frac-neg 34.2%

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

   10. distribute-frac-neg 34.2%

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

   11. unpow2 34.2%

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

   12. associate--l+ 45.7%

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

   13. metadata-eval 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.7%

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

   15. unpow2 45.7%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in k around 0 28.7%

   \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. +-commutative 28.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]

   2. fma-def 28.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]

   3. unpow2 28.7%

      \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   4. *-commutative 28.7%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   5. times-frac 29.5%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   6. times-frac 32.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right)}\right) \]

   7. associate-*r* 32.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}}\right) \]

   8. unpow2 32.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right) \]

   9. unpow2 32.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right) \]

   10. times-frac 36.8%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right) \]

   11. distribute-rgt-out 36.8%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \frac{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}{{t}^{2}}\right) \]

   12. metadata-eval 36.8%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \frac{t \cdot \color{blue}{0.16666666666666666}}{{t}^{2}}\right) \]

   13. metadata-eval 36.8%

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

   14. unpow2 36.8%

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

   15. times-frac 43.1%

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

   16. metadata-eval 43.1%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

6. Simplified 43.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)} \]

7. Taylor expanded in t around 0 37.1%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
8. Step-by-step derivation

   1. fma-def 37.1%

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

   2. unpow2 37.1%

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

   3. unpow2 37.1%

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

   4. times-frac 39.3%

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

   5. unpow2 39.3%

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

   6. unpow2 39.3%

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

   7. associate-*r/ 46.6%

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

9. Simplified 46.6%

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

10. Taylor expanded in l around 0 41.9%

    \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}\right) \cdot {\ell}^{2}}{t}} \]
11. Step-by-step derivation

    1. associate-/l* 41.8%

       \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}}{\frac{t}{{\ell}^{2}}}} \]

    2. associate-*r/ 41.8%

       \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{{k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}}{\frac{t}{{\ell}^{2}}} \]

    3. metadata-eval 41.8%

       \[\leadsto \frac{\frac{\color{blue}{2}}{{k}^{4}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2}}}{\frac{t}{{\ell}^{2}}} \]

    4. associate-*r/ 41.8%

       \[\leadsto \frac{\frac{2}{{k}^{4}} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2}}}}{\frac{t}{{\ell}^{2}}} \]

    5. metadata-eval 41.8%

       \[\leadsto \frac{\frac{2}{{k}^{4}} - \frac{\color{blue}{0.3333333333333333}}{{k}^{2}}}{\frac{t}{{\ell}^{2}}} \]

    6. unpow2 41.8%

       \[\leadsto \frac{\frac{2}{{k}^{4}} - \frac{0.3333333333333333}{\color{blue}{k \cdot k}}}{\frac{t}{{\ell}^{2}}} \]

    7. unpow2 41.8%

       \[\leadsto \frac{\frac{2}{{k}^{4}} - \frac{0.3333333333333333}{k \cdot k}}{\frac{t}{\color{blue}{\ell \cdot \ell}}} \]

12. Simplified 41.8%

    \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{4}} - \frac{0.3333333333333333}{k \cdot k}}{\frac{t}{\ell \cdot \ell}}} \]

13. Taylor expanded in k around inf 30.8%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. *-commutative 30.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot -0.3333333333333333} \]

    2. unpow2 30.8%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \cdot -0.3333333333333333 \]

    3. unpow2 30.8%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot -0.3333333333333333 \]

    4. associate-*r* 31.5%

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot -0.3333333333333333 \]

15. Simplified 31.5%

    \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333} \]

16. Final simplification 31.5%

    \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333 \]

Reproduce

herbie shell --seed 2023279 
(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))))