Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.3% → 91.1%

Time: 34.6s

Alternatives: 8

Speedup: 38.3×

• 35.1% 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: 34.3% 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: 91.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. associate-/r* 37.0%

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

   2. *-commutative 37.0%

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

   3. associate-/r* 40.8%

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

   4. associate-*r/ 41.9%

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

   5. associate-/l* 40.8%

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

   6. +-commutative 40.8%

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

   7. unpow2 40.8%

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

   8. sqr-neg 40.8%

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

   9. distribute-frac-neg 40.8%

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

   10. distribute-frac-neg 40.8%

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

   11. unpow2 40.8%

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

   12. associate--l+ 49.1%

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

   13. metadata-eval 49.1%

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

   14. +-rgt-identity 49.1%

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

   15. unpow2 49.1%

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

   16. distribute-frac-neg 49.1%

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

3. Simplified 49.1%

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

4. Taylor expanded in k around inf 73.0%

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

   1. associate-/r* 73.3%

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

   2. associate-/r* 73.6%

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

   3. unpow2 73.6%

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

   4. times-frac 77.5%

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

   5. unpow2 77.5%

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

6. Simplified 77.5%

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

7. Taylor expanded in l around 0 73.6%

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

   1. times-frac 73.6%

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

   2. unpow2 73.6%

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

   3. unpow2 73.6%

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

   4. times-frac 94.2%

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

   5. unpow2 94.2%

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

9. Simplified 94.2%

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

    1. unpow2 94.2%

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

11. Applied egg-rr 94.2%

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

12. Final simplification 94.2%

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

Alternative 2: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;k \leq 1.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot t_2}}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{t_2}{t}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)) (t_2 (pow (/ l k) 2.0)))
   (if (<= k 1.5e-63)
     (/ 2.0 (/ (* k (* k t)) (* (cos k) t_2)))
     (if (<= k 3.2e+152)
       (* 2.0 (* (/ (* l l) (* k k)) (/ (cos k) (* t t_1))))
       (* 2.0 (/ (/ t_2 t) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = pow((l / k), 2.0);
	double tmp;
	if (k <= 1.5e-63) {
		tmp = 2.0 / ((k * (k * t)) / (cos(k) * t_2));
	} else if (k <= 3.2e+152) {
		tmp = 2.0 * (((l * l) / (k * k)) * (cos(k) / (t * t_1)));
	} else {
		tmp = 2.0 * ((t_2 / t) / t_1);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = (l / k) ** 2.0d0
    if (k <= 1.5d-63) then
        tmp = 2.0d0 / ((k * (k * t)) / (cos(k) * t_2))
    else if (k <= 3.2d+152) then
        tmp = 2.0d0 * (((l * l) / (k * k)) * (cos(k) / (t * t_1)))
    else
        tmp = 2.0d0 * ((t_2 / t) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = Math.pow((l / k), 2.0);
	double tmp;
	if (k <= 1.5e-63) {
		tmp = 2.0 / ((k * (k * t)) / (Math.cos(k) * t_2));
	} else if (k <= 3.2e+152) {
		tmp = 2.0 * (((l * l) / (k * k)) * (Math.cos(k) / (t * t_1)));
	} else {
		tmp = 2.0 * ((t_2 / t) / t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	t_2 = math.pow((l / k), 2.0)
	tmp = 0
	if k <= 1.5e-63:
		tmp = 2.0 / ((k * (k * t)) / (math.cos(k) * t_2))
	elif k <= 3.2e+152:
		tmp = 2.0 * (((l * l) / (k * k)) * (math.cos(k) / (t * t_1)))
	else:
		tmp = 2.0 * ((t_2 / t) / t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(l / k) ^ 2.0
	tmp = 0.0
	if (k <= 1.5e-63)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * t)) / Float64(cos(k) * t_2)));
	elseif (k <= 3.2e+152)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(cos(k) / Float64(t * t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(t_2 / t) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	t_2 = (l / k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1.5e-63)
		tmp = 2.0 / ((k * (k * t)) / (cos(k) * t_2));
	elseif (k <= 3.2e+152)
		tmp = 2.0 * (((l * l) / (k * k)) * (cos(k) / (t * t_1)));
	else
		tmp = 2.0 * ((t_2 / t) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.5e-63], N[(2.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.2e+152], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$2 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.4999999999999999e-63

   1. Initial program 38.8%

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

   3. Applied egg-rr 16.9%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   4. Step-by-step derivation

      1. unpow2 16.9%

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

      2. pow-sqr 16.9%

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

      3. *-commutative 16.9%

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

      4. *-commutative 16.9%

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

      5. associate-*l* 16.9%

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

      6. metadata-eval 16.9%

         \[\leadsto \frac{2}{\sqrt{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{\color{blue}{4}}}} \]

   5. Simplified 16.9%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{4}}}} \]

   6. Taylor expanded in k around inf 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-/l* 73.2%

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

      3. associate-/l* 73.2%

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

      4. associate-/r/ 73.2%

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

      5. unpow2 73.2%

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

      6. unpow2 73.2%

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

      7. times-frac 91.9%

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

      8. unpow2 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in k around 0 77.1%

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

       1. unpow2 77.1%

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

       2. associate-*l* 79.0%

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

   11. Simplified 79.0%

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

   if 1.4999999999999999e-63 < k < 3.20000000000000005e152

   1. Initial program 28.4%

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

      1. associate-/r* 27.9%

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

      2. *-commutative 27.9%

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

      3. associate-/r* 32.3%

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

      4. associate-*r/ 32.2%

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

      5. associate-/l* 32.3%

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

      6. +-commutative 32.3%

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

      7. unpow2 32.3%

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

      8. sqr-neg 32.3%

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

      9. distribute-frac-neg 32.3%

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

      10. distribute-frac-neg 32.3%

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

      11. unpow2 32.3%

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

      12. associate--l+ 41.4%

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

      13. metadata-eval 41.4%

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

      14. +-rgt-identity 41.4%

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

      15. unpow2 41.4%

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

      16. distribute-frac-neg 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in k around inf 86.6%

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

      1. associate-/r* 89.0%

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

      2. associate-/r* 89.0%

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

      3. unpow2 89.0%

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

      4. times-frac 89.1%

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

      5. unpow2 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in l around 0 86.6%

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

      1. times-frac 89.0%

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

      2. unpow2 89.0%

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

      3. unpow2 89.0%

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

      4. times-frac 93.0%

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

      5. unpow2 93.0%

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

   9. Simplified 93.0%

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

       1. unpow2 93.0%

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

   11. Applied egg-rr 93.0%

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

   12. Taylor expanded in l around 0 86.6%

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

       1. times-frac 89.0%

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

       2. unpow2 89.0%

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

       3. unpow2 89.0%

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

   14. Simplified 89.0%

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

   if 3.20000000000000005e152 < k

   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* 39.1%

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

      2. *-commutative 39.1%

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

      3. associate-/r* 43.4%

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

      4. associate-*r/ 43.4%

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

      5. associate-/l* 43.4%

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

      6. +-commutative 43.4%

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

      7. unpow2 43.4%

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

      8. sqr-neg 43.4%

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

      9. distribute-frac-neg 43.4%

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

      10. distribute-frac-neg 43.4%

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

      11. unpow2 43.4%

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

      12. associate--l+ 53.8%

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

      13. metadata-eval 53.8%

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

      14. +-rgt-identity 53.8%

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

      15. unpow2 53.8%

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

      16. distribute-frac-neg 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in k around inf 59.6%

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

      1. associate-/r* 59.6%

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

      2. associate-/r* 59.6%

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

      3. unpow2 59.6%

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

      4. times-frac 68.5%

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

      5. unpow2 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in k around 0 59.6%

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

      1. unpow2 59.6%

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

      2. unpow2 59.6%

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

      3. times-frac 63.6%

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

      4. unpow2 63.6%

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

   9. Simplified 63.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 77.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 1.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \ell}{\frac{{\sin k}^{2} \cdot t_1}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 1.5e-63)
     (/ 2.0 (/ t_1 (* (cos k) (pow (/ l k) 2.0))))
     (* 2.0 (/ (* l l) (/ (* (pow (sin k) 2.0) t_1) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 1.5e-63) {
		tmp = 2.0 / (t_1 / (cos(k) * pow((l / k), 2.0)));
	} else {
		tmp = 2.0 * ((l * l) / ((pow(sin(k), 2.0) * t_1) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k * t)
    if (k <= 1.5d-63) then
        tmp = 2.0d0 / (t_1 / (cos(k) * ((l / k) ** 2.0d0)))
    else
        tmp = 2.0d0 * ((l * l) / (((sin(k) ** 2.0d0) * t_1) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 1.5e-63) {
		tmp = 2.0 / (t_1 / (Math.cos(k) * Math.pow((l / k), 2.0)));
	} else {
		tmp = 2.0 * ((l * l) / ((Math.pow(Math.sin(k), 2.0) * t_1) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 1.5e-63:
		tmp = 2.0 / (t_1 / (math.cos(k) * math.pow((l / k), 2.0)))
	else:
		tmp = 2.0 * ((l * l) / ((math.pow(math.sin(k), 2.0) * t_1) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 1.5e-63)
		tmp = Float64(2.0 / Float64(t_1 / Float64(cos(k) * (Float64(l / k) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * l) / Float64(Float64((sin(k) ^ 2.0) * t_1) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 1.5e-63)
		tmp = 2.0 / (t_1 / (cos(k) * ((l / k) ^ 2.0)));
	else
		tmp = 2.0 * ((l * l) / (((sin(k) ^ 2.0) * t_1) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.5e-63], N[(2.0 / N[(t$95$1 / N[(N[Cos[k], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4999999999999999e-63

   1. Initial program 38.8%

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

   3. Applied egg-rr 16.9%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   4. Step-by-step derivation

      1. unpow2 16.9%

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

      2. pow-sqr 16.9%

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

      3. *-commutative 16.9%

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

      4. *-commutative 16.9%

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

      5. associate-*l* 16.9%

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

      6. metadata-eval 16.9%

         \[\leadsto \frac{2}{\sqrt{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{\color{blue}{4}}}} \]

   5. Simplified 16.9%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{4}}}} \]

   6. Taylor expanded in k around inf 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-/l* 73.2%

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

      3. associate-/l* 73.2%

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

      4. associate-/r/ 73.2%

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

      5. unpow2 73.2%

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

      6. unpow2 73.2%

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

      7. times-frac 91.9%

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

      8. unpow2 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in k around 0 77.1%

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

       1. unpow2 77.1%

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

       2. associate-*l* 79.0%

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

   11. Simplified 79.0%

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

   if 1.4999999999999999e-63 < k

   1. Initial program 34.0%

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

      1. associate-/r* 33.8%

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

      2. *-commutative 33.8%

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

      3. associate-/r* 38.1%

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

      4. associate-*r/ 38.1%

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

      5. associate-/l* 38.1%

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

      6. +-commutative 38.1%

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

      7. unpow2 38.1%

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

      8. sqr-neg 38.1%

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

      9. distribute-frac-neg 38.1%

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

      10. distribute-frac-neg 38.1%

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

      11. unpow2 38.1%

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

      12. associate--l+ 47.9%

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

      13. metadata-eval 47.9%

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

      14. +-rgt-identity 47.9%

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

      15. unpow2 47.9%

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

      16. distribute-frac-neg 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 72.4%

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

      1. associate-/r* 73.5%

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

      2. associate-/r* 73.5%

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

      3. unpow2 73.5%

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

      4. times-frac 78.2%

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

      5. unpow2 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in l around 0 72.4%

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

      1. times-frac 73.5%

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

      2. unpow2 73.5%

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

      3. unpow2 73.5%

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

      4. times-frac 95.5%

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

      5. unpow2 95.5%

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

   9. Simplified 95.5%

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

   10. Taylor expanded in l around 0 72.4%

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

       1. associate-/l* 72.4%

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

       2. unpow2 72.4%

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

       3. associate-*r* 72.4%

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

       4. unpow2 72.4%

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

       5. associate-*r* 76.7%

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

   12. Simplified 76.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \ell}{\frac{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\cos k}}\\ \end{array} \]

Alternative 4: 72.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{t_1}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ l k) 2.0)))
   (if (<= t -7.5e-171)
     (/ 2.0 (/ (* k (* k t)) (* (cos k) t_1)))
     (* 2.0 (/ (/ t_1 t) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((l / k), 2.0);
	double tmp;
	if (t <= -7.5e-171) {
		tmp = 2.0 / ((k * (k * t)) / (cos(k) * t_1));
	} else {
		tmp = 2.0 * ((t_1 / t) / pow(sin(k), 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) :: t_1
    real(8) :: tmp
    t_1 = (l / k) ** 2.0d0
    if (t <= (-7.5d-171)) then
        tmp = 2.0d0 / ((k * (k * t)) / (cos(k) * t_1))
    else
        tmp = 2.0d0 * ((t_1 / t) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((l / k), 2.0);
	double tmp;
	if (t <= -7.5e-171) {
		tmp = 2.0 / ((k * (k * t)) / (Math.cos(k) * t_1));
	} else {
		tmp = 2.0 * ((t_1 / t) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((l / k), 2.0)
	tmp = 0
	if t <= -7.5e-171:
		tmp = 2.0 / ((k * (k * t)) / (math.cos(k) * t_1))
	else:
		tmp = 2.0 * ((t_1 / t) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / k) ^ 2.0
	tmp = 0.0
	if (t <= -7.5e-171)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * t)) / Float64(cos(k) * t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 / t) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) ^ 2.0;
	tmp = 0.0;
	if (t <= -7.5e-171)
		tmp = 2.0 / ((k * (k * t)) / (cos(k) * t_1));
	else
		tmp = 2.0 * ((t_1 / t) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -7.5e-171], N[(2.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000033e-171

   1. Initial program 47.2%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   4. Step-by-step derivation

      1. unpow2 0.0%

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

      2. pow-sqr 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*l* 0.0%

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

      6. metadata-eval 0.0%

         \[\leadsto \frac{2}{\sqrt{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{\color{blue}{4}}}} \]

   5. Simplified 0.0%

      \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{4}}}} \]

   6. Taylor expanded in k around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-/l* 77.4%

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

      3. associate-/l* 77.5%

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

      4. associate-/r/ 77.5%

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

      5. unpow2 77.5%

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

      6. unpow2 77.5%

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

      7. times-frac 94.3%

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

      8. unpow2 94.3%

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

   8. Simplified 94.3%

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

   9. Taylor expanded in k around 0 75.8%

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

       1. unpow2 75.8%

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

       2. associate-*l* 78.3%

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

   11. Simplified 78.3%

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

   if -7.50000000000000033e-171 < t

   1. Initial program 28.7%

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

      1. associate-/r* 28.7%

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

      2. *-commutative 28.7%

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

      3. associate-/r* 31.8%

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

      4. associate-*r/ 33.2%

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

      5. associate-/l* 31.8%

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

      6. +-commutative 31.8%

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

      7. unpow2 31.8%

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

      8. sqr-neg 31.8%

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

      9. distribute-frac-neg 31.8%

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

      10. distribute-frac-neg 31.8%

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

      11. unpow2 31.8%

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

      12. associate--l+ 37.7%

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

      13. metadata-eval 37.7%

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

      14. +-rgt-identity 37.7%

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

      15. unpow2 37.7%

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

      16. distribute-frac-neg 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in k around inf 71.1%

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

      1. associate-/r* 69.9%

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

      2. associate-/r* 70.0%

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

      3. unpow2 70.0%

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

      4. times-frac 75.2%

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

      5. unpow2 75.2%

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

   6. Simplified 75.2%

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

   7. Taylor expanded in k around 0 62.0%

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

      1. unpow2 62.0%

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

      2. unpow2 62.0%

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

      3. times-frac 72.7%

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

      4. unpow2 72.7%

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

   9. Simplified 72.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 5: 72.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. associate-/r* 37.0%

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

   2. *-commutative 37.0%

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

   3. associate-/r* 40.8%

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

   4. associate-*r/ 41.9%

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

   5. associate-/l* 40.8%

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

   6. +-commutative 40.8%

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

   7. unpow2 40.8%

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

   8. sqr-neg 40.8%

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

   9. distribute-frac-neg 40.8%

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

   10. distribute-frac-neg 40.8%

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

   11. unpow2 40.8%

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

   12. associate--l+ 49.1%

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

   13. metadata-eval 49.1%

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

   14. +-rgt-identity 49.1%

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

   15. unpow2 49.1%

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

   16. distribute-frac-neg 49.1%

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

3. Simplified 49.1%

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

4. Taylor expanded in k around inf 73.0%

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

   1. associate-/r* 73.3%

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

   2. associate-/r* 73.6%

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

   3. unpow2 73.6%

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

   4. times-frac 77.5%

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

   5. unpow2 77.5%

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

6. Simplified 77.5%

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

7. Taylor expanded in l around 0 73.6%

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

   1. times-frac 73.6%

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

   2. unpow2 73.6%

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

   3. unpow2 73.6%

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

   4. times-frac 94.2%

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

   5. unpow2 94.2%

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

9. Simplified 94.2%

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

    1. unpow2 94.2%

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

11. Applied egg-rr 94.2%

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

12. Taylor expanded in k around 0 72.9%

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

    1. unpow2 72.9%

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

14. Simplified 72.9%

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

15. Final simplification 72.9%

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

Alternative 6: 67.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.3e+42)
   (* 2.0 (* (/ l (pow k 4.0)) (/ l t)))
   (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e+42) {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / t));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d+42) then
        tmp = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t))
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e+42) {
		tmp = 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.3e+42:
		tmp = 2.0 * ((l / math.pow(k, 4.0)) * (l / t))
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.3e+42)
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.3e+42)
		tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.3e+42], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.29999999999999995e42

   1. Initial program 38.0%

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

      1. associate-/r* 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-/r* 41.4%

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

      4. associate-*r/ 43.0%

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

      5. associate-/l* 41.4%

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

      6. +-commutative 41.4%

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

      7. unpow2 41.4%

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

      8. sqr-neg 41.4%

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

      9. distribute-frac-neg 41.4%

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

      10. distribute-frac-neg 41.4%

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

      11. unpow2 41.4%

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

      12. associate--l+ 49.0%

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

      13. metadata-eval 49.0%

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

      14. +-rgt-identity 49.0%

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

      15. unpow2 49.0%

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

      16. distribute-frac-neg 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in k around inf 75.6%

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

      1. associate-/r* 75.5%

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

      2. associate-/r* 75.8%

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

      3. unpow2 75.8%

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

      4. times-frac 78.8%

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

      5. unpow2 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in k around 0 64.6%

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

      1. unpow2 64.6%

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

      2. *-commutative 64.6%

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

      3. times-frac 68.9%

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

   9. Simplified 68.9%

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

   if 1.29999999999999995e42 < k

   1. Initial program 34.8%

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

      1. associate-/r* 34.8%

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

      2. *-commutative 34.8%

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

      3. associate-/r* 39.1%

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

      4. associate-*r/ 39.1%

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

      5. associate-/l* 39.2%

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

      6. +-commutative 39.2%

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

      7. unpow2 39.2%

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

      8. sqr-neg 39.2%

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

      9. distribute-frac-neg 39.2%

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

      10. distribute-frac-neg 39.2%

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

      11. unpow2 39.2%

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

      12. associate--l+ 49.3%

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

      13. metadata-eval 49.3%

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

      14. +-rgt-identity 49.3%

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

      15. unpow2 49.3%

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

      16. distribute-frac-neg 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in k around 0 54.7%

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

      1. fma-def 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. associate-*l* 54.9%

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

      5. associate-*r/ 54.9%

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

      6. times-frac 54.0%

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

      7. unpow2 54.0%

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

      8. associate-/l* 54.0%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in k around inf 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-/r* 54.7%

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

      3. unpow2 54.7%

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

      4. unpow2 54.7%

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

      5. times-frac 57.2%

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

      6. unpow2 57.2%

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

   9. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 7: 68.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.3e+42)
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
   (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e+42) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.3d+42) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.3e+42) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.3e+42:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.3e+42)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.3e+42)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.3e+42], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.29999999999999995e42

   1. Initial program 38.0%

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

      1. associate-/r* 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-/r* 41.4%

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

      4. associate-*r/ 43.0%

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

      5. associate-/l* 41.4%

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

      6. +-commutative 41.4%

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

      7. unpow2 41.4%

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

      8. sqr-neg 41.4%

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

      9. distribute-frac-neg 41.4%

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

      10. distribute-frac-neg 41.4%

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

      11. unpow2 41.4%

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

      12. associate--l+ 49.0%

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

      13. metadata-eval 49.0%

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

      14. +-rgt-identity 49.0%

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

      15. unpow2 49.0%

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

      16. distribute-frac-neg 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in k around inf 75.6%

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

      1. associate-/r* 75.5%

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

      2. associate-/r* 75.8%

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

      3. unpow2 75.8%

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

      4. times-frac 78.8%

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

      5. unpow2 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in k around 0 64.6%

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

      1. unpow2 64.6%

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

      2. *-commutative 64.6%

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

      3. times-frac 68.9%

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

   9. Simplified 68.9%

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

       1. associate-*l/ 71.4%

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

   11. Applied egg-rr 71.4%

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

   if 1.29999999999999995e42 < k

   1. Initial program 34.8%

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

      1. associate-/r* 34.8%

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

      2. *-commutative 34.8%

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

      3. associate-/r* 39.1%

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

      4. associate-*r/ 39.1%

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

      5. associate-/l* 39.2%

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

      6. +-commutative 39.2%

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

      7. unpow2 39.2%

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

      8. sqr-neg 39.2%

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

      9. distribute-frac-neg 39.2%

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

      10. distribute-frac-neg 39.2%

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

      11. unpow2 39.2%

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

      12. associate--l+ 49.3%

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

      13. metadata-eval 49.3%

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

      14. +-rgt-identity 49.3%

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

      15. unpow2 49.3%

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

      16. distribute-frac-neg 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in k around 0 54.7%

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

      1. fma-def 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. associate-*l* 54.9%

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

      5. associate-*r/ 54.9%

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

      6. times-frac 54.0%

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

      7. unpow2 54.0%

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

      8. associate-/l* 54.0%

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

   6. Simplified 54.0%

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

   7. Taylor expanded in k around inf 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-/r* 54.7%

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

      3. unpow2 54.7%

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

      4. unpow2 54.7%

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

      5. times-frac 57.2%

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

      6. unpow2 57.2%

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

   9. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+42}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 8: 34.4% accurate, 38.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (* (/ l k) (/ l (* k t)))))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * ((l / k) * (l / (k * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(-0.3333333333333333 * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.1%

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

   1. associate-/r* 37.0%

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

   2. *-commutative 37.0%

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

   3. associate-/r* 40.8%

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

   4. associate-*r/ 41.9%

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

   5. associate-/l* 40.8%

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

   6. +-commutative 40.8%

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

   7. unpow2 40.8%

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

   8. sqr-neg 40.8%

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

   9. distribute-frac-neg 40.8%

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

   10. distribute-frac-neg 40.8%

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

   11. unpow2 40.8%

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

   12. associate--l+ 49.1%

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

   13. metadata-eval 49.1%

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

   14. +-rgt-identity 49.1%

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

   15. unpow2 49.1%

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

   16. distribute-frac-neg 49.1%

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

3. Simplified 49.1%

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

4. Taylor expanded in k around 0 39.4%

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

   1. fma-def 39.4%

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

   2. unpow2 39.4%

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

   3. unpow2 39.4%

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

   4. associate-*l* 41.1%

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

   5. associate-*r/ 41.1%

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

   6. times-frac 39.0%

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

   7. unpow2 39.0%

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

   8. associate-/l* 39.0%

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

6. Simplified 39.0%

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

7. Taylor expanded in k around inf 37.1%

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

   1. *-commutative 37.1%

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

   2. unpow2 37.1%

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

   3. unpow2 37.1%

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

   4. associate-*r* 37.4%

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

9. Simplified 37.4%

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

    1. times-frac 38.2%

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

11. Applied egg-rr 38.2%

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

12. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2023285 
(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))))