Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 99.4%

Time: 23.0s

Alternatives: 10

Speedup: 2.0×

• 34.9% 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: 35.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \frac{k}{\ell}\\ \frac{\sqrt{2}}{t_1} \cdot \frac{\sqrt{2}}{\frac{t_1 \cdot t}{\cos k}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * (k / l);
	return (Math.sqrt(2.0) / t_1) * (Math.sqrt(2.0) / ((t_1 * t) / Math.cos(k)));
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * (k / l)
	return (math.sqrt(2.0) / t_1) * (math.sqrt(2.0) / ((t_1 * t) / math.cos(k)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

   1. add-sqr-sqrt 15.8%

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

   2. pow2 15.8%

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

4. Applied egg-rr 23.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 40.5%

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

   1. clear-num 40.5%

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

   2. inv-pow 40.5%

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

   3. *-commutative 40.5%

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

   4. unpow-prod-down 37.5%

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

   5. pow2 37.5%

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

   6. add-sqr-sqrt 90.6%

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

   7. associate-/l* 92.0%

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

7. Applied egg-rr 92.0%

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

   1. unpow-1 92.0%

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

   2. associate-/l* 92.0%

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

   3. associate-/r/ 92.1%

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

9. Simplified 92.1%

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

    1. clear-num 92.3%

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

    2. add-sqr-sqrt 92.3%

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

    3. *-un-lft-identity 92.3%

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

    4. times-frac 92.2%

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

    5. sqrt-div 92.2%

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

    6. associate-/r/ 92.2%

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

    7. unpow2 92.2%

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

    8. sqrt-prod 55.5%

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

    9. add-sqr-sqrt 57.4%

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

    10. div-inv 57.4%

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

    11. clear-num 57.4%

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

11. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k \cdot \frac{\sin k}{\ell}}}{1} \cdot \frac{\frac{\sqrt{2}}{k \cdot \frac{\sin k}{\ell}}}{\frac{t}{\cos k}}} \]
12. Step-by-step derivation

    1. /-rgt-identity 99.4%

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

    2. associate-*r/ 97.1%

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

    3. associate-*l/ 99.5%

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

    4. *-commutative 99.5%

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

    5. associate-/r* 99.4%

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

    6. associate-*r/ 99.5%

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

    7. associate-*r/ 97.2%

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

    8. associate-*l/ 99.5%

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

    9. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * (k / l);
	return (Math.sqrt(2.0) / (t_1 * t)) * ((Math.sqrt(2.0) / t_1) * Math.cos(k));
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * (k / l)
	return (math.sqrt(2.0) / (t_1 * t)) * ((math.sqrt(2.0) / t_1) * math.cos(k))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

   1. add-sqr-sqrt 15.8%

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

   2. pow2 15.8%

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

4. Applied egg-rr 23.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 40.5%

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

   1. clear-num 40.5%

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

   2. inv-pow 40.5%

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

   3. *-commutative 40.5%

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

   4. unpow-prod-down 37.5%

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

   5. pow2 37.5%

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

   6. add-sqr-sqrt 90.6%

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

   7. associate-/l* 92.0%

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

7. Applied egg-rr 92.0%

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

   1. unpow-1 92.0%

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

   2. associate-/l* 92.0%

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

   3. associate-/r/ 92.1%

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

9. Simplified 92.1%

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

    1. clear-num 92.3%

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

    2. add-sqr-sqrt 92.3%

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

    3. div-inv 92.3%

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

    4. times-frac 92.2%

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

    5. sqrt-div 92.2%

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

    6. associate-/r/ 92.3%

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

    7. unpow2 92.3%

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

    8. sqrt-prod 55.5%

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

    9. add-sqr-sqrt 57.4%

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

    10. div-inv 57.4%

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

    11. clear-num 57.4%

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

11. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{k \cdot \frac{\sin k}{\ell}}}{t} \cdot \frac{\frac{\sqrt{2}}{k \cdot \frac{\sin k}{\ell}}}{\frac{1}{\cos k}}} \]
12. Step-by-step derivation

    1. associate-/l/ 99.4%

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

    2. associate-*r/ 97.1%

       \[\leadsto \frac{\sqrt{2}}{t \cdot \color{blue}{\frac{k \cdot \sin k}{\ell}}} \cdot \frac{\frac{\sqrt{2}}{k \cdot \frac{\sin k}{\ell}}}{\frac{1}{\cos k}} \]

    3. associate-*l/ 99.5%

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

    4. *-commutative 99.5%

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

    5. associate-/r/ 99.5%

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

    6. /-rgt-identity 99.5%

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

    7. associate-*r/ 97.2%

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

    8. associate-*l/ 99.5%

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

    9. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t, l, k):
	t_1 = k * (math.sin(k) / l)
	return 2.0 / (t_1 * ((t / math.cos(k)) * t_1))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

   1. add-sqr-sqrt 15.8%

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

   2. pow2 15.8%

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

4. Applied egg-rr 23.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 40.5%

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

   1. associate-/l* 41.1%

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

7. Simplified 41.1%

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

   1. unpow2 41.1%

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

   2. *-commutative 41.1%

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

   3. *-commutative 41.1%

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

   4. swap-sqr 37.9%

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

   5. add-sqr-sqrt 92.0%

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

   6. associate-*r* 99.2%

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

   7. div-inv 99.2%

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

   8. clear-num 99.3%

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

   9. div-inv 99.2%

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

   10. clear-num 99.2%

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 4: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

   1. add-sqr-sqrt 15.8%

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

   2. pow2 15.8%

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

4. Applied egg-rr 23.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 40.5%

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

   1. clear-num 40.5%

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

   2. inv-pow 40.5%

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

   3. *-commutative 40.5%

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

   4. unpow-prod-down 37.5%

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

   5. pow2 37.5%

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

   6. add-sqr-sqrt 90.6%

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

   7. associate-/l* 92.0%

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

7. Applied egg-rr 92.0%

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

   1. unpow-1 92.0%

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

   2. associate-/l* 92.0%

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

   3. associate-/r/ 92.1%

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

9. Simplified 92.1%

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

    1. clear-num 92.3%

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

    2. div-inv 92.3%

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

    3. *-un-lft-identity 92.3%

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

    4. times-frac 92.3%

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

    5. metadata-eval 92.3%

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

    6. associate-/r/ 92.3%

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

    7. pow-flip 92.3%

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

    8. div-inv 92.3%

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

    9. clear-num 92.3%

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

    10. metadata-eval 92.3%

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

11. Applied egg-rr 92.3%

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

    1. associate-/r/ 92.3%

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

    2. associate-*r/ 90.8%

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

    3. associate-*l/ 92.3%

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

    4. *-commutative 92.3%

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

13. Simplified 92.3%

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

14. Final simplification 92.3%

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

Alternative 5: 29.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 * (((l / sqrt(t)) * (k ** (-2.0d0))) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.2%

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

   5. +-commutative 38.2%

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

   6. unpow2 38.2%

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

   7. sqr-neg 38.2%

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

   8. distribute-frac-neg 38.2%

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

   9. distribute-frac-neg 38.2%

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

   10. unpow2 38.2%

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

   11. associate--l+ 43.6%

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

   12. metadata-eval 43.6%

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

   13. +-rgt-identity 43.6%

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

   14. unpow2 43.6%

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

   15. distribute-frac-neg 43.6%

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

   16. distribute-frac-neg 43.6%

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

3. Simplified 43.6%

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

5. Taylor expanded in k around inf 73.5%

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

   1. times-frac 75.2%

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

7. Simplified 75.2%

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

8. Taylor expanded in k around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-/r* 60.2%

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

10. Simplified 60.2%

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

    1. div-inv 60.2%

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

    2. pow-flip 60.2%

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

    3. metadata-eval 60.2%

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

12. Applied egg-rr 60.2%

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

    1. add-sqr-sqrt 40.4%

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

    2. pow2 40.4%

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

    3. sqrt-prod 34.7%

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

    4. sqrt-div 28.4%

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

    5. unpow2 28.4%

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

    6. sqrt-prod 13.1%

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

    7. add-sqr-sqrt 29.5%

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

    8. sqrt-pow1 30.2%

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

    9. metadata-eval 30.2%

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

14. Applied egg-rr 30.2%

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

15. Final simplification 30.2%

    \[\leadsto 2 \cdot {\left(\frac{\ell}{\sqrt{t}} \cdot {k}^{-2}\right)}^{2} \]
16. Add Preprocessing

Alternative 6: 29.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 / ((((k ** 2.0d0) / l) * sqrt(t)) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

   1. add-sqr-sqrt 15.8%

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

   2. pow2 15.8%

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

4. Applied egg-rr 23.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

5. Taylor expanded in k around 0 30.3%

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

6. Final simplification 30.3%

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

Alternative 7: 26.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{k}^{-4}}{t} \cdot {\ell}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.6e+173)
   (/ 2.0 (pow (* (/ k t) (* k (/ (pow t 1.5) l))) 2.0))
   (* 2.0 (* (/ (pow k -4.0) t) (pow l 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.6e+173) {
		tmp = 2.0 / pow(((k / t) * (k * (pow(t, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 * ((pow(k, -4.0) / t) * pow(l, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 5.6d+173) then
        tmp = 2.0d0 / (((k / t) * (k * ((t ** 1.5d0) / l))) ** 2.0d0)
    else
        tmp = 2.0d0 * (((k ** (-4.0d0)) / t) * (l ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.6e+173) {
		tmp = 2.0 / Math.pow(((k / t) * (k * (Math.pow(t, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(k, -4.0) / t) * Math.pow(l, 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.6e+173:
		tmp = 2.0 / math.pow(((k / t) * (k * (math.pow(t, 1.5) / l))), 2.0)
	else:
		tmp = 2.0 * ((math.pow(k, -4.0) / t) * math.pow(l, 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.6e+173)
		tmp = Float64(2.0 / (Float64(Float64(k / t) * Float64(k * Float64((t ^ 1.5) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((k ^ -4.0) / t) * (l ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.6e+173)
		tmp = 2.0 / (((k / t) * (k * ((t ^ 1.5) / l))) ^ 2.0);
	else
		tmp = 2.0 * (((k ^ -4.0) / t) * (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.6e+173], N[(2.0 / N[Power[N[(N[(k / t), $MachinePrecision] * N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[k, -4.0], $MachinePrecision] / t), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.59999999999999964e173

   1. Initial program 40.4%

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

      1. add-sqr-sqrt 16.8%

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

      2. pow2 16.8%

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

   4. Applied egg-rr 21.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

   5. Taylor expanded in k around 0 24.8%

      \[\leadsto \frac{2}{{\left(\frac{k}{t} \cdot \left(\color{blue}{k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}} \]

   if 5.59999999999999964e173 < t

   1. Initial program 9.5%

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

      1. associate-/r* 9.5%

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

      2. *-commutative 9.5%

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

      3. associate-*l* 9.5%

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

      4. associate-*l/ 9.5%

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

      5. +-commutative 9.5%

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

      6. unpow2 9.5%

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

      7. sqr-neg 9.5%

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

      8. distribute-frac-neg 9.5%

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

      9. distribute-frac-neg 9.5%

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

      10. unpow2 9.5%

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

      11. associate--l+ 29.4%

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

      12. metadata-eval 29.4%

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

      13. +-rgt-identity 29.4%

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

      14. unpow2 29.4%

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

      15. distribute-frac-neg 29.4%

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

      16. distribute-frac-neg 29.4%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in k around inf 82.1%

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

      1. times-frac 92.2%

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

   7. Simplified 92.2%

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

   8. Taylor expanded in k around 0 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-/r* 67.8%

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

   10. Simplified 67.8%

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

       1. div-inv 67.8%

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

       2. pow-flip 67.8%

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

       3. metadata-eval 67.8%

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

   12. Applied egg-rr 67.8%

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

       1. expm1-log1p-u 67.8%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-udef 67.8%

          \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]

       3. *-commutative 67.8%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-4} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

       4. clear-num 67.8%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left({k}^{-4} \cdot \color{blue}{\frac{1}{\frac{t}{{\ell}^{2}}}}\right)} - 1\right) \]

       5. un-div-inv 67.8%

          \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{k}^{-4}}{\frac{t}{{\ell}^{2}}}}\right)} - 1\right) \]

   14. Applied egg-rr 67.8%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{k}^{-4}}{\frac{t}{{\ell}^{2}}}\right)} - 1\right)} \]
   15. Step-by-step derivation

       1. expm1-def 67.8%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{-4}}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

       2. expm1-log1p 67.8%

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

       3. associate-/r/ 77.7%

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

   16. Simplified 77.7%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{k}^{-4}}{t} \cdot {\ell}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{{k}^{-4}}{t} \cdot {\ell}^{2}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ (pow k -4.0) t) (pow l 2.0))))

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(k, -4.0) / t) * pow(l, 2.0));
}

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 * (((k ** (-4.0d0)) / t) * (l ** 2.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((Math.pow(k, -4.0) / t) * Math.pow(l, 2.0));
}

Python

def code(t, l, k):
	return 2.0 * ((math.pow(k, -4.0) / t) * math.pow(l, 2.0))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64((k ^ -4.0) / t) * (l ^ 2.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((k ^ -4.0) / t) * (l ^ 2.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[k, -4.0], $MachinePrecision] / t), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.2%

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

   5. +-commutative 38.2%

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

   6. unpow2 38.2%

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

   7. sqr-neg 38.2%

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

   8. distribute-frac-neg 38.2%

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

   9. distribute-frac-neg 38.2%

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

   10. unpow2 38.2%

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

   11. associate--l+ 43.6%

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

   12. metadata-eval 43.6%

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

   13. +-rgt-identity 43.6%

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

   14. unpow2 43.6%

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

   15. distribute-frac-neg 43.6%

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

   16. distribute-frac-neg 43.6%

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

3. Simplified 43.6%

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

5. Taylor expanded in k around inf 73.5%

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

   1. times-frac 75.2%

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

7. Simplified 75.2%

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

8. Taylor expanded in k around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-/r* 60.2%

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

10. Simplified 60.2%

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

    1. div-inv 60.2%

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

    2. pow-flip 60.2%

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

    3. metadata-eval 60.2%

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

12. Applied egg-rr 60.2%

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

    1. expm1-log1p-u 40.5%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

    2. expm1-udef 39.7%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]

    3. *-commutative 39.7%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{-4} \cdot \frac{{\ell}^{2}}{t}}\right)} - 1\right) \]

    4. clear-num 39.7%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left({k}^{-4} \cdot \color{blue}{\frac{1}{\frac{t}{{\ell}^{2}}}}\right)} - 1\right) \]

    5. un-div-inv 39.7%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{k}^{-4}}{\frac{t}{{\ell}^{2}}}}\right)} - 1\right) \]

14. Applied egg-rr 39.7%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{k}^{-4}}{\frac{t}{{\ell}^{2}}}\right)} - 1\right)} \]
15. Step-by-step derivation

    1. expm1-def 40.5%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{-4}}{\frac{t}{{\ell}^{2}}}\right)\right)} \]

    2. expm1-log1p 60.2%

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

    3. associate-/r/ 62.7%

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

16. Simplified 62.7%

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

17. Final simplification 62.7%

    \[\leadsto 2 \cdot \left(\frac{{k}^{-4}}{t} \cdot {\ell}^{2}\right) \]
18. Add Preprocessing

Alternative 9: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ (* (pow k -4.0) (pow l 2.0)) t)))

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(k, -4.0) * pow(l, 2.0)) / 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 = 2.0d0 * (((k ** (-4.0d0)) * (l ** 2.0d0)) / t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((Math.pow(k, -4.0) * Math.pow(l, 2.0)) / t);
}

Python

def code(t, l, k):
	return 2.0 * ((math.pow(k, -4.0) * math.pow(l, 2.0)) / t)

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64((k ^ -4.0) * (l ^ 2.0)) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((k ^ -4.0) * (l ^ 2.0)) / t);
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[k, -4.0], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.8%

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

   4. associate-*l/ 38.2%

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

   5. +-commutative 38.2%

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

   6. unpow2 38.2%

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

   7. sqr-neg 38.2%

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

   8. distribute-frac-neg 38.2%

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

   9. distribute-frac-neg 38.2%

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

   10. unpow2 38.2%

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

   11. associate--l+ 43.6%

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

   12. metadata-eval 43.6%

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

   13. +-rgt-identity 43.6%

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

   14. unpow2 43.6%

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

   15. distribute-frac-neg 43.6%

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

   16. distribute-frac-neg 43.6%

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

3. Simplified 43.6%

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

5. Taylor expanded in k around inf 73.5%

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

   1. times-frac 75.2%

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

7. Simplified 75.2%

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

8. Taylor expanded in k around 0 62.7%

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

   1. *-commutative 62.7%

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

   2. associate-/r* 60.2%

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

10. Simplified 60.2%

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

    1. div-inv 60.2%

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

    2. pow-flip 60.2%

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

    3. metadata-eval 60.2%

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

12. Applied egg-rr 60.2%

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

    1. associate-*l/ 62.7%

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

14. Applied egg-rr 62.7%

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

15. Final simplification 62.7%

    \[\leadsto 2 \cdot \frac{{k}^{-4} \cdot {\ell}^{2}}{t} \]
16. Add Preprocessing

Alternative 10: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* t (/ (pow k 4.0) (pow l 2.0)))))

C

double code(double t, double l, double k) {
	return 2.0 / (t * (pow(k, 4.0) / pow(l, 2.0)));
}

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 * ((k ** 4.0d0) / (l ** 2.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (t * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
}

Python

def code(t, l, k):
	return 2.0 / (t * (math.pow(k, 4.0) / math.pow(l, 2.0)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(t * Float64((k ^ 4.0) / (l ^ 2.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (t * ((k ^ 4.0) / (l ^ 2.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. add-sqr-sqrt 15.8%

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

   2. pow2 15.8%

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

4. Applied egg-rr 23.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}} \]

5. Taylor expanded in k around inf 40.5%

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

6. Taylor expanded in k around 0 62.7%

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

   1. associate-*l/ 62.7%

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

8. Simplified 62.7%

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

9. Final simplification 62.7%

   \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))