Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 82.5%

Time: 17.2s

Alternatives: 13

Speedup: 3.8×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{1}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        2e+229)
     (/ 2.0 (* (* (/ (pow t 3.0) l) (/ (sin k) l)) (* (tan k) t_1)))
     (/
      2.0
      (/ (* (* t (/ 1.0 (pow (/ l k) 2.0))) (pow (sin k) 2.0)) (cos k))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 2e+229) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (sin(k) / l)) * (tan(k) * t_1));
	} else {
		tmp = 2.0 / (((t * (1.0 / pow((l / k), 2.0))) * pow(sin(k), 2.0)) / 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 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 2d+229) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (sin(k) / l)) * (tan(k) * t_1))
    else
        tmp = 2.0d0 / (((t * (1.0d0 / ((l / k) ** 2.0d0))) * (sin(k) ** 2.0d0)) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1)) <= 2e+229) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (Math.sin(k) / l)) * (Math.tan(k) * t_1));
	} else {
		tmp = 2.0 / (((t * (1.0 / Math.pow((l / k), 2.0))) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * t_1)) <= 2e+229:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (math.sin(k) / l)) * (math.tan(k) * t_1))
	else:
		tmp = 2.0 / (((t * (1.0 / math.pow((l / k), 2.0))) * math.pow(math.sin(k), 2.0)) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1)) <= 2e+229)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(sin(k) / l)) * Float64(tan(k) * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(1.0 / (Float64(l / k) ^ 2.0))) * (sin(k) ^ 2.0)) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 2e+229)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (sin(k) / l)) * (tan(k) * t_1));
	else
		tmp = 2.0 / (((t * (1.0 / ((l / k) ^ 2.0))) * (sin(k) ^ 2.0)) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] * t$95$1), $MachinePrecision]), $MachinePrecision], 2e+229], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(1.0 / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2e229

   1. Initial program 78.0%

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

      1. associate-*l* 78.0%

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

      2. +-commutative 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in t around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. times-frac 87.2%

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

   6. Simplified 87.2%

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

   if 2e229 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.7%

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

      1. associate-*l* 20.7%

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

      2. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in t around 0 60.3%

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

      1. times-frac 58.5%

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

      2. unpow2 58.5%

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

      3. *-commutative 58.5%

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

      4. unpow2 58.5%

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

      5. times-frac 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in k around inf 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*r* 60.3%

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

      3. *-commutative 60.3%

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

      4. times-frac 60.3%

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

      5. *-commutative 60.3%

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

      6. associate-/l* 59.4%

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

      7. unpow2 59.4%

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

      8. unpow2 59.4%

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

      9. times-frac 81.8%

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

   9. Simplified 81.8%

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

       1. associate-*r/ 81.8%

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

       2. pow2 81.8%

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

   11. Applied egg-rr 81.8%

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

       1. div-inv 81.8%

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

   13. Applied egg-rr 81.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{1}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 2: 81.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{1}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1)))))
   (if (<= t_2 (- INFINITY))
     (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l)))))
     (if (<= t_2 INFINITY)
       (* l (* l (/ 2.0 (* (tan k) (* (+ 2.0 t_1) (* (pow t 3.0) (sin k)))))))
       (/
        2.0
        (/ (* (* t (/ 1.0 (pow (/ l k) 2.0))) (pow (sin k) 2.0)) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = (((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = l * (l * (2.0 / (tan(k) * ((2.0 + t_1) * (pow(t, 3.0) * sin(k))))));
	} else {
		tmp = 2.0 / (((t * (1.0 / pow((l / k), 2.0))) * pow(sin(k), 2.0)) / cos(k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = (((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = l * (l * (2.0 / (Math.tan(k) * ((2.0 + t_1) * (Math.pow(t, 3.0) * Math.sin(k))))));
	} else {
		tmp = 2.0 / (((t * (1.0 / Math.pow((l / k), 2.0))) * Math.pow(Math.sin(k), 2.0)) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	t_2 = (((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	elif t_2 <= math.inf:
		tmp = l * (l * (2.0 / (math.tan(k) * ((2.0 + t_1) * (math.pow(t, 3.0) * math.sin(k))))))
	else:
		tmp = 2.0 / (((t * (1.0 / math.pow((l / k), 2.0))) * math.pow(math.sin(k), 2.0)) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	elseif (t_2 <= Inf)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + t_1) * Float64((t ^ 3.0) * sin(k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(1.0 / (Float64(l / k) ^ 2.0))) * (sin(k) ^ 2.0)) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	t_2 = ((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	elseif (t_2 <= Inf)
		tmp = l * (l * (2.0 / (tan(k) * ((2.0 + t_1) * ((t ^ 3.0) * sin(k))))));
	else
		tmp = 2.0 / (((t * (1.0 / ((l / k) ^ 2.0))) * (sin(k) ^ 2.0)) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(2.0 / N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(1.0 / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < -inf.0

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. +-commutative 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in k around 0 69.9%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 68.7%

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

      2. unpow2 68.7%

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

      3. unpow2 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in k around 0 69.9%

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 70.9%

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

      2. unpow2 70.9%

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

      3. unpow2 70.9%

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

      4. *-commutative 70.9%

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

      5. times-frac 86.2%

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

   9. Simplified 86.2%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l/ 83.5%

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

      3. associate-*l/ 83.0%

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

      4. associate-/r/ 83.1%

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

      5. *-commutative 83.1%

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

      6. associate-/l/ 83.1%

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

      7. associate-*r* 83.6%

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

      8. *-commutative 83.6%

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

      9. associate-*r* 83.6%

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

      10. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. expm1-log1p-u 55.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 50.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 51.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 58.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 87.5%

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

      3. *-commutative 87.5%

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

   7. Simplified 87.5%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. +-commutative 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in t around 0 48.7%

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

      1. times-frac 46.4%

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

      2. unpow2 46.4%

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

      3. *-commutative 46.4%

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

      4. unpow2 46.4%

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

      5. times-frac 56.6%

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

   6. Simplified 56.6%

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

   7. Taylor expanded in k around inf 48.7%

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

      1. *-commutative 48.7%

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

      2. associate-*r* 48.7%

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

      3. *-commutative 48.7%

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

      4. times-frac 48.7%

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

      5. *-commutative 48.7%

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

      6. associate-/l* 47.6%

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

      7. unpow2 47.6%

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

      8. unpow2 47.6%

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

      9. times-frac 76.5%

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

   9. Simplified 76.5%

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

       1. associate-*r/ 76.5%

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

       2. pow2 76.5%

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

   11. Applied egg-rr 76.5%

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

       1. div-inv 76.5%

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

   13. Applied egg-rr 76.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq -\infty:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{1}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 3: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9e-30)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (if (<= k 1.02e+149)
     (* l (* l (/ 2.0 (* (tan k) (* t (* (sin k) (* k k)))))))
     (* (cos k) (/ 2.0 (* (pow (sin k) 2.0) (/ t (pow (/ l k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e-30) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (k <= 1.02e+149) {
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	} else {
		tmp = cos(k) * (2.0 / (pow(sin(k), 2.0) * (t / pow((l / 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) :: tmp
    if (k <= 9d-30) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (k <= 1.02d+149) then
        tmp = l * (l * (2.0d0 / (tan(k) * (t * (sin(k) * (k * k))))))
    else
        tmp = cos(k) * (2.0d0 / ((sin(k) ** 2.0d0) * (t / ((l / k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e-30) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (k <= 1.02e+149) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (t * (Math.sin(k) * (k * k))))));
	} else {
		tmp = Math.cos(k) * (2.0 / (Math.pow(Math.sin(k), 2.0) * (t / Math.pow((l / k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9e-30:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif k <= 1.02e+149:
		tmp = l * (l * (2.0 / (math.tan(k) * (t * (math.sin(k) * (k * k))))))
	else:
		tmp = math.cos(k) * (2.0 / (math.pow(math.sin(k), 2.0) * (t / math.pow((l / k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9e-30)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (k <= 1.02e+149)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(t * Float64(sin(k) * Float64(k * k)))))));
	else
		tmp = Float64(cos(k) * Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(t / (Float64(l / k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9e-30)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (k <= 1.02e+149)
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	else
		tmp = cos(k) * (2.0 / ((sin(k) ^ 2.0) * (t / ((l / k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9e-30], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.02e+149], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.99999999999999935e-30

   1. Initial program 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-*l/ 57.6%

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

      3. associate-*l/ 57.2%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

      6. associate-/l/ 57.2%

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

      7. associate-*r* 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.4%

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

      10. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around 0 56.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 66.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   if 8.99999999999999935e-30 < k < 1.01999999999999997e149

   1. Initial program 56.3%

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

      1. associate-/l/ 56.3%

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

      2. associate-*l/ 56.3%

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

      3. associate-*l/ 56.2%

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

      4. associate-/r/ 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-/l/ 56.7%

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

      7. associate-*r* 57.0%

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

      8. *-commutative 57.0%

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

      9. associate-*r* 57.0%

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

      10. *-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 49.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 52.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 60.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 60.6%

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

      3. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in k around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. unpow2 77.1%

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

   10. Simplified 77.1%

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

   if 1.01999999999999997e149 < k

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. +-commutative 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in t around 0 67.1%

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

      1. times-frac 67.0%

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

      2. unpow2 67.0%

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

      3. *-commutative 67.0%

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

      4. unpow2 67.0%

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

      5. times-frac 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in k around inf 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*r* 67.1%

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

      3. *-commutative 67.1%

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

      4. times-frac 67.1%

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

      5. *-commutative 67.1%

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

      6. associate-/l* 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*r/ 94.0%

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

       2. pow2 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. associate-/r/ 94.0%

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

   13. Applied egg-rr 94.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 4: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.8e-30)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (if (<= k 1.8e+149)
     (* l (* l (/ 2.0 (* (tan k) (* t (* (sin k) (* k k)))))))
     (/ 2.0 (/ (* (pow (sin k) 2.0) (/ t (pow (/ l k) 2.0))) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.8e-30) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (k <= 1.8e+149) {
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t / pow((l / k), 2.0))) / 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) :: tmp
    if (k <= 8.8d-30) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (k <= 1.8d+149) then
        tmp = l * (l * (2.0d0 / (tan(k) * (t * (sin(k) * (k * k))))))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * (t / ((l / k) ** 2.0d0))) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.8e-30) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (k <= 1.8e+149) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (t * (Math.sin(k) * (k * k))))));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t / Math.pow((l / k), 2.0))) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.8e-30:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif k <= 1.8e+149:
		tmp = l * (l * (2.0 / (math.tan(k) * (t * (math.sin(k) * (k * k))))))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * (t / math.pow((l / k), 2.0))) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.8e-30)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (k <= 1.8e+149)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(t * Float64(sin(k) * Float64(k * k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t / (Float64(l / k) ^ 2.0))) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.8e-30)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (k <= 1.8e+149)
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) * (t / ((l / k) ^ 2.0))) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.8e-30], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e+149], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.79999999999999933e-30

   1. Initial program 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-*l/ 57.6%

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

      3. associate-*l/ 57.2%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

      6. associate-/l/ 57.2%

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

      7. associate-*r* 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.4%

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

      10. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around 0 56.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 66.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   if 8.79999999999999933e-30 < k < 1.79999999999999997e149

   1. Initial program 56.3%

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

      1. associate-/l/ 56.3%

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

      2. associate-*l/ 56.3%

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

      3. associate-*l/ 56.2%

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

      4. associate-/r/ 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-/l/ 56.7%

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

      7. associate-*r* 57.0%

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

      8. *-commutative 57.0%

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

      9. associate-*r* 57.0%

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

      10. *-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 49.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 52.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 60.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 60.6%

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

      3. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in k around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. unpow2 77.1%

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

   10. Simplified 77.1%

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

   if 1.79999999999999997e149 < k

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. +-commutative 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in t around 0 67.1%

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

      1. times-frac 67.0%

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

      2. unpow2 67.0%

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

      3. *-commutative 67.0%

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

      4. unpow2 67.0%

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

      5. times-frac 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in k around inf 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*r* 67.1%

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

      3. *-commutative 67.1%

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

      4. times-frac 67.1%

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

      5. *-commutative 67.1%

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

      6. associate-/l* 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*r/ 94.0%

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

       2. pow2 94.0%

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

   11. Applied egg-rr 94.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 5: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-30)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (if (<= k 1.7e+149)
     (* l (* l (/ 2.0 (* (tan k) (* t (* (sin k) (* k k)))))))
     (/ 2.0 (/ (/ (* t (pow (sin k) 2.0)) (pow (/ l k) 2.0)) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-30) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (k <= 1.7e+149) {
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	} else {
		tmp = 2.0 / (((t * pow(sin(k), 2.0)) / pow((l / k), 2.0)) / 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) :: tmp
    if (k <= 8.5d-30) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (k <= 1.7d+149) then
        tmp = l * (l * (2.0d0 / (tan(k) * (t * (sin(k) * (k * k))))))
    else
        tmp = 2.0d0 / (((t * (sin(k) ** 2.0d0)) / ((l / k) ** 2.0d0)) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-30) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (k <= 1.7e+149) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (t * (Math.sin(k) * (k * k))))));
	} else {
		tmp = 2.0 / (((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / k), 2.0)) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.5e-30:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif k <= 1.7e+149:
		tmp = l * (l * (2.0 / (math.tan(k) * (t * (math.sin(k) * (k * k))))))
	else:
		tmp = 2.0 / (((t * math.pow(math.sin(k), 2.0)) / math.pow((l / k), 2.0)) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-30)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (k <= 1.7e+149)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(t * Float64(sin(k) * Float64(k * k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / k) ^ 2.0)) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.5e-30)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (k <= 1.7e+149)
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	else
		tmp = 2.0 / (((t * (sin(k) ^ 2.0)) / ((l / k) ^ 2.0)) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.5e-30], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e+149], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.49999999999999931e-30

   1. Initial program 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-*l/ 57.6%

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

      3. associate-*l/ 57.2%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

      6. associate-/l/ 57.2%

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

      7. associate-*r* 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.4%

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

      10. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around 0 56.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 66.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   if 8.49999999999999931e-30 < k < 1.6999999999999999e149

   1. Initial program 56.3%

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

      1. associate-/l/ 56.3%

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

      2. associate-*l/ 56.3%

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

      3. associate-*l/ 56.2%

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

      4. associate-/r/ 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-/l/ 56.7%

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

      7. associate-*r* 57.0%

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

      8. *-commutative 57.0%

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

      9. associate-*r* 57.0%

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

      10. *-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 49.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 52.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 60.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 60.6%

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

      3. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in k around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. unpow2 77.1%

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

   10. Simplified 77.1%

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

   if 1.6999999999999999e149 < k

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. +-commutative 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in t around 0 67.1%

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

      1. times-frac 67.0%

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

      2. unpow2 67.0%

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

      3. *-commutative 67.0%

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

      4. unpow2 67.0%

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

      5. times-frac 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in k around inf 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*r* 67.1%

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

      3. *-commutative 67.1%

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

      4. times-frac 67.1%

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

      5. *-commutative 67.1%

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

      6. associate-/l* 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.9%

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

   9. Simplified 93.9%

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

       1. associate-*r/ 94.0%

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

       2. pow2 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. associate-*l/ 94.1%

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

   13. Applied egg-rr 94.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 6: 70.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-30)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (if (<= k 2.3e+149)
     (* l (* l (/ 2.0 (* (tan k) (* t (* (sin k) (* k k)))))))
     (/ 2.0 (* (/ t (* (/ l k) (/ l k))) (/ (pow (sin k) 2.0) (cos k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-30) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else if (k <= 2.3e+149) {
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	} else {
		tmp = 2.0 / ((t / ((l / k) * (l / k))) * (pow(sin(k), 2.0) / 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) :: tmp
    if (k <= 9.5d-30) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else if (k <= 2.3d+149) then
        tmp = l * (l * (2.0d0 / (tan(k) * (t * (sin(k) * (k * k))))))
    else
        tmp = 2.0d0 / ((t / ((l / k) * (l / k))) * ((sin(k) ** 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-30) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else if (k <= 2.3e+149) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (t * (Math.sin(k) * (k * k))))));
	} else {
		tmp = 2.0 / ((t / ((l / k) * (l / k))) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.5e-30:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	elif k <= 2.3e+149:
		tmp = l * (l * (2.0 / (math.tan(k) * (t * (math.sin(k) * (k * k))))))
	else:
		tmp = 2.0 / ((t / ((l / k) * (l / k))) * (math.pow(math.sin(k), 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-30)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	elseif (k <= 2.3e+149)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(t * Float64(sin(k) * Float64(k * k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / Float64(Float64(l / k) * Float64(l / k))) * Float64((sin(k) ^ 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-30)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	elseif (k <= 2.3e+149)
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	else
		tmp = 2.0 / ((t / ((l / k) * (l / k))) * ((sin(k) ^ 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.5e-30], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.3e+149], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.49999999999999939e-30

   1. Initial program 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-*l/ 57.6%

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

      3. associate-*l/ 57.2%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

      6. associate-/l/ 57.2%

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

      7. associate-*r* 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.4%

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

      10. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around 0 56.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 66.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   if 9.49999999999999939e-30 < k < 2.2999999999999998e149

   1. Initial program 56.3%

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

      1. associate-/l/ 56.3%

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

      2. associate-*l/ 56.3%

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

      3. associate-*l/ 56.2%

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

      4. associate-/r/ 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-/l/ 56.7%

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

      7. associate-*r* 57.0%

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

      8. *-commutative 57.0%

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

      9. associate-*r* 57.0%

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

      10. *-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 49.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 52.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 60.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 60.6%

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

      3. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in k around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. unpow2 77.1%

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

   10. Simplified 77.1%

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

   if 2.2999999999999998e149 < k

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. +-commutative 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in t around 0 67.1%

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

      1. times-frac 67.0%

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

      2. unpow2 67.0%

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

      3. *-commutative 67.0%

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

      4. unpow2 67.0%

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

      5. times-frac 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in k around inf 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*r* 67.1%

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

      3. *-commutative 67.1%

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

      4. times-frac 67.1%

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

      5. *-commutative 67.1%

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

      6. associate-/l* 67.1%

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

      7. unpow2 67.1%

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

      8. unpow2 67.1%

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

      9. times-frac 93.9%

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

   9. Simplified 93.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+149}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 7: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-30)
   (* (/ l k) (/ (/ l (pow t 3.0)) k))
   (* l (* l (/ 2.0 (* (tan k) (* t (* (sin k) (* k k)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-30) {
		tmp = (l / k) * ((l / pow(t, 3.0)) / k);
	} else {
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * 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) :: tmp
    if (k <= 7d-30) then
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    else
        tmp = l * (l * (2.0d0 / (tan(k) * (t * (sin(k) * (k * k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-30) {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	} else {
		tmp = l * (l * (2.0 / (Math.tan(k) * (t * (Math.sin(k) * (k * k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7e-30:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	else:
		tmp = l * (l * (2.0 / (math.tan(k) * (t * (math.sin(k) * (k * k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-30)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(t * Float64(sin(k) * Float64(k * k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7e-30)
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	else
		tmp = l * (l * (2.0 / (tan(k) * (t * (sin(k) * (k * k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7e-30], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000006e-30

   1. Initial program 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-*l/ 57.6%

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

      3. associate-*l/ 57.2%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

      6. associate-/l/ 57.2%

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

      7. associate-*r* 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.4%

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

      10. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around 0 56.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 56.4%

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

      2. *-commutative 56.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 61.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 66.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   if 7.0000000000000006e-30 < k

   1. Initial program 51.0%

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

      1. associate-/l/ 51.0%

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

      2. associate-*l/ 51.0%

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

      3. associate-*l/ 51.0%

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

      4. associate-/r/ 51.3%

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

      5. *-commutative 51.3%

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

      6. associate-/l/ 51.2%

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

      7. associate-*r* 51.4%

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

      8. *-commutative 51.4%

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

      9. associate-*r* 51.4%

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

      10. *-commutative 51.4%

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

   3. Simplified 51.4%

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

      1. expm1-log1p-u 51.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 47.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 49.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 53.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 53.7%

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

      3. *-commutative 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in k around inf 72.7%

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

      1. associate-*r* 72.7%

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

      2. unpow2 72.7%

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

   10. Simplified 72.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\right)\\ \end{array} \]

Alternative 8: 70.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-53}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.5e-53)
   (* l (/ (/ l k) (* (pow t 3.0) k)))
   (if (<= t 1.8e-76)
     (/ 2.0 (* (/ (* k k) (cos k)) (* (/ t l) (/ k (/ l k)))))
     (* (/ l k) (/ (/ l (pow t 3.0)) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-53) {
		tmp = l * ((l / k) / (pow(t, 3.0) * k));
	} else if (t <= 1.8e-76) {
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * (k / (l / k))));
	} else {
		tmp = (l / k) * ((l / pow(t, 3.0)) / 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) :: tmp
    if (t <= (-3.5d-53)) then
        tmp = l * ((l / k) / ((t ** 3.0d0) * k))
    else if (t <= 1.8d-76) then
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((t / l) * (k / (l / k))))
    else
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-53) {
		tmp = l * ((l / k) / (Math.pow(t, 3.0) * k));
	} else if (t <= 1.8e-76) {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((t / l) * (k / (l / k))));
	} else {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.5e-53:
		tmp = l * ((l / k) / (math.pow(t, 3.0) * k))
	elif t <= 1.8e-76:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((t / l) * (k / (l / k))))
	else:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.5e-53)
		tmp = Float64(l * Float64(Float64(l / k) / Float64((t ^ 3.0) * k)));
	elseif (t <= 1.8e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(t / l) * Float64(k / Float64(l / k)))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.5e-53)
		tmp = l * ((l / k) / ((t ^ 3.0) * k));
	elseif (t <= 1.8e-76)
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * (k / (l / k))));
	else
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.5e-53], N[(l * N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e-76], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.49999999999999993e-53

   1. Initial program 62.1%

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

      1. associate-/l/ 62.1%

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

      2. associate-*l/ 63.3%

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

      3. associate-*l/ 63.1%

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

      4. associate-/r/ 63.0%

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

      5. *-commutative 63.0%

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

      6. associate-/l/ 63.0%

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

      7. associate-*r* 63.0%

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

      8. *-commutative 63.0%

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

      9. associate-*r* 63.0%

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

      10. *-commutative 63.0%

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

   3. Simplified 63.0%

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

      1. expm1-log1p-u 50.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 46.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 47.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 52.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.4%

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

      3. *-commutative 66.4%

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

   7. Simplified 66.4%

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

   8. Taylor expanded in k around 0 64.0%

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

   9. Taylor expanded in k around 0 61.6%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   10. Step-by-step derivation

       1. associate-/r* 61.5%

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

       2. unpow2 61.5%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

       3. associate-/r* 67.4%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       4. associate-/l/ 70.2%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

       5. *-commutative 70.2%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{k}}{\color{blue}{k \cdot {t}^{3}}} \]

   11. Simplified 70.2%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k \cdot {t}^{3}}} \]

   if -3.49999999999999993e-53 < t < 1.8e-76

   1. Initial program 39.9%

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

      1. associate-*l* 39.9%

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

      2. +-commutative 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around 0 80.5%

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

      1. times-frac 77.5%

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

      2. unpow2 77.5%

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

      3. *-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. times-frac 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in k around 0 79.1%

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

      1. unpow2 79.1%

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

      2. associate-/l* 79.1%

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

   9. Simplified 79.1%

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

   if 1.8e-76 < t

   1. Initial program 66.5%

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

      1. associate-/l/ 66.5%

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

      2. associate-*l/ 67.6%

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

      3. associate-*l/ 67.5%

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

      4. associate-/r/ 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-/l/ 67.5%

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

      7. associate-*r* 67.6%

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

      8. *-commutative 67.6%

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

      9. associate-*r* 67.5%

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

      10. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in k around 0 58.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.4%

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

      2. *-commutative 58.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 59.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 68.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-53}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \end{array} \]

Alternative 9: 67.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-53}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-76}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.8e-53)
   (* l (/ (/ l k) (* (pow t 3.0) k)))
   (if (<= t 1.9e-76)
     (* (* 2.0 l) (/ (/ l t) (pow k 4.0)))
     (* (/ l k) (/ (/ l (pow t 3.0)) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.8e-53) {
		tmp = l * ((l / k) / (pow(t, 3.0) * k));
	} else if (t <= 1.9e-76) {
		tmp = (2.0 * l) * ((l / t) / pow(k, 4.0));
	} else {
		tmp = (l / k) * ((l / pow(t, 3.0)) / 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) :: tmp
    if (t <= (-2.8d-53)) then
        tmp = l * ((l / k) / ((t ** 3.0d0) * k))
    else if (t <= 1.9d-76) then
        tmp = (2.0d0 * l) * ((l / t) / (k ** 4.0d0))
    else
        tmp = (l / k) * ((l / (t ** 3.0d0)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.8e-53) {
		tmp = l * ((l / k) / (Math.pow(t, 3.0) * k));
	} else if (t <= 1.9e-76) {
		tmp = (2.0 * l) * ((l / t) / Math.pow(k, 4.0));
	} else {
		tmp = (l / k) * ((l / Math.pow(t, 3.0)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2.8e-53:
		tmp = l * ((l / k) / (math.pow(t, 3.0) * k))
	elif t <= 1.9e-76:
		tmp = (2.0 * l) * ((l / t) / math.pow(k, 4.0))
	else:
		tmp = (l / k) * ((l / math.pow(t, 3.0)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.8e-53)
		tmp = Float64(l * Float64(Float64(l / k) / Float64((t ^ 3.0) * k)));
	elseif (t <= 1.9e-76)
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(l / t) / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / (t ^ 3.0)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2.8e-53)
		tmp = l * ((l / k) / ((t ^ 3.0) * k));
	elseif (t <= 1.9e-76)
		tmp = (2.0 * l) * ((l / t) / (k ^ 4.0));
	else
		tmp = (l / k) * ((l / (t ^ 3.0)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.8e-53], N[(l * N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-76], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.79999999999999985e-53

   1. Initial program 62.1%

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

      1. associate-/l/ 62.1%

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

      2. associate-*l/ 63.3%

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

      3. associate-*l/ 63.1%

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

      4. associate-/r/ 63.0%

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

      5. *-commutative 63.0%

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

      6. associate-/l/ 63.0%

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

      7. associate-*r* 63.0%

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

      8. *-commutative 63.0%

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

      9. associate-*r* 63.0%

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

      10. *-commutative 63.0%

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

   3. Simplified 63.0%

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

      1. expm1-log1p-u 50.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 46.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 47.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 52.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 66.4%

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

      3. *-commutative 66.4%

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

   7. Simplified 66.4%

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

   8. Taylor expanded in k around 0 64.0%

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

   9. Taylor expanded in k around 0 61.6%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   10. Step-by-step derivation

       1. associate-/r* 61.5%

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

       2. unpow2 61.5%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

       3. associate-/r* 67.4%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       4. associate-/l/ 70.2%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

       5. *-commutative 70.2%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{k}}{\color{blue}{k \cdot {t}^{3}}} \]

   11. Simplified 70.2%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k \cdot {t}^{3}}} \]

   if -2.79999999999999985e-53 < t < 1.9000000000000001e-76

   1. Initial program 39.9%

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

      1. associate-*l* 39.9%

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

      2. +-commutative 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around 0 80.5%

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

      1. times-frac 77.5%

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

      2. unpow2 77.5%

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

      3. *-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. times-frac 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in k around inf 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*r* 80.5%

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

      3. *-commutative 80.5%

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

      4. times-frac 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-/l* 79.5%

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

      7. unpow2 79.5%

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

      8. unpow2 79.5%

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

      9. times-frac 92.9%

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

   9. Simplified 92.9%

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

       1. associate-*r/ 92.9%

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

       2. pow2 92.9%

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

   11. Applied egg-rr 92.9%

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

   12. Taylor expanded in k around 0 66.5%

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

       1. unpow2 66.5%

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

       2. *-commutative 66.5%

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

       3. associate-*r/ 75.7%

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

       4. associate-*r* 75.7%

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

       5. associate-/r* 76.6%

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

   14. Simplified 76.6%

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

   if 1.9000000000000001e-76 < t

   1. Initial program 66.5%

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

      1. associate-/l/ 66.5%

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

      2. associate-*l/ 67.6%

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

      3. associate-*l/ 67.5%

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

      4. associate-/r/ 67.6%

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

      5. *-commutative 67.6%

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

      6. associate-/l/ 67.5%

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

      7. associate-*r* 67.6%

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

      8. *-commutative 67.6%

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

      9. associate-*r* 67.5%

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

      10. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in k around 0 58.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 58.4%

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

      2. *-commutative 58.4%

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

      3. times-frac 61.3%

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

      4. unpow2 61.3%

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   6. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 59.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]

   8. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}} \]
   9. Step-by-step derivation

      1. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]

   10. Applied egg-rr 68.3%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-53}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-76}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{{t}^{3}}}{k}\\ \end{array} \]

Alternative 10: 62.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7e-30)
   (* l (/ (/ l k) (* (pow t 3.0) k)))
   (* 2.0 (/ l (/ (* t (pow k 4.0)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-30) {
		tmp = l * ((l / k) / (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * (l / ((t * pow(k, 4.0)) / l));
	}
	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 <= 7d-30) then
        tmp = l * ((l / k) / ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 * (l / ((t * (k ** 4.0d0)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7e-30) {
		tmp = l * ((l / k) / (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * (l / ((t * Math.pow(k, 4.0)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7e-30:
		tmp = l * ((l / k) / (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 * (l / ((t * math.pow(k, 4.0)) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7e-30)
		tmp = Float64(l * Float64(Float64(l / k) / Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 * Float64(l / Float64(Float64(t * (k ^ 4.0)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7e-30)
		tmp = l * ((l / k) / ((t ^ 3.0) * k));
	else
		tmp = 2.0 * (l / ((t * (k ^ 4.0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7e-30], N[(l * N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l / N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000006e-30

   1. Initial program 56.6%

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

      1. associate-/l/ 56.6%

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

      2. associate-*l/ 57.6%

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

      3. associate-*l/ 57.2%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

      6. associate-/l/ 57.2%

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

      7. associate-*r* 57.4%

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

      8. *-commutative 57.4%

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

      9. associate-*r* 57.4%

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

      10. *-commutative 57.4%

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

   3. Simplified 57.4%

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

      1. expm1-log1p-u 39.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 37.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 39.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

   5. Applied egg-rr 39.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 43.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 64.0%

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

      3. *-commutative 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in k around 0 64.1%

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

   9. Taylor expanded in k around 0 61.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]
   10. Step-by-step derivation

       1. associate-/r* 61.3%

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

       2. unpow2 61.3%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

       3. associate-/r* 65.9%

          \[\leadsto \ell \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k}}}{{t}^{3}} \]

       4. associate-/l/ 66.9%

          \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3} \cdot k}} \]

       5. *-commutative 66.9%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{k}}{\color{blue}{k \cdot {t}^{3}}} \]

   11. Simplified 66.9%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k \cdot {t}^{3}}} \]

   if 7.0000000000000006e-30 < k

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in t around 0 69.8%

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

      1. times-frac 69.8%

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

      2. unpow2 69.8%

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

      3. *-commutative 69.8%

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

      4. unpow2 69.8%

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

      5. times-frac 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in k around 0 58.7%

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

      1. unpow2 58.7%

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

      2. associate-/l* 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}}\\ \end{array} \]

Alternative 11: 55.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. associate-*l* 55.1%

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

   2. +-commutative 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in t around 0 61.4%

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

   1. times-frac 60.7%

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

   2. unpow2 60.7%

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

   3. *-commutative 60.7%

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

   4. unpow2 60.7%

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

   5. times-frac 66.2%

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

6. Simplified 66.2%

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

7. Taylor expanded in k around 0 54.9%

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

   1. unpow2 54.9%

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

   2. associate-/l* 59.6%

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

9. Simplified 59.6%

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

    1. associate-/r/ 59.6%

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

    2. *-commutative 59.6%

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

11. Applied egg-rr 59.6%

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

12. Final simplification 59.6%

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

Alternative 12: 55.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l / ((t * (k ** 4.0d0)) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. associate-*l* 55.1%

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

   2. +-commutative 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in t around 0 61.4%

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

   1. times-frac 60.7%

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

   2. unpow2 60.7%

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

   3. *-commutative 60.7%

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

   4. unpow2 60.7%

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

   5. times-frac 66.2%

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

6. Simplified 66.2%

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

7. Taylor expanded in k around 0 54.9%

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

   1. unpow2 54.9%

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

   2. associate-/l* 59.6%

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

9. Simplified 59.6%

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

10. Final simplification 59.6%

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

Alternative 13: 55.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l / ((k ** 4.0d0) / (l / t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. associate-*l* 55.1%

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

   2. +-commutative 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in t around 0 61.4%

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

   1. times-frac 60.7%

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

   2. unpow2 60.7%

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

   3. *-commutative 60.7%

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

   4. unpow2 60.7%

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

   5. times-frac 66.2%

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

6. Simplified 66.2%

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

7. Taylor expanded in k around 0 54.9%

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

   1. unpow2 54.9%

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

   2. associate-/l* 59.6%

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

9. Simplified 59.6%

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

    1. expm1-log1p-u 42.9%

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

    2. expm1-udef 30.2%

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

    3. *-commutative 30.2%

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

11. Applied egg-rr 30.2%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 59.6%

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

    3. *-commutative 59.6%

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

    4. associate-/l* 59.9%

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

13. Simplified 59.9%

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

14. Final simplification 59.9%

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

Reproduce

herbie shell --seed 2023192 
(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))))