Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 93.1%

Time: 8.2s

Alternatives: 14

Speedup: 4.4×

• 38.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 36.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    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: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{\sin k\_m \cdot \left(\tan k\_m \cdot t\right)}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell \cdot 2}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.2e+160)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (sin k_m) (* (tan k_m) t)) l))) l))
   (*
    (/ (* (cos k_m) l) k_m)
    (/ (* l 2.0) (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e+160) {
		tmp = 2.0 / ((k_m * (k_m * ((sin(k_m) * (tan(k_m) * t)) / l))) / l);
	} else {
		tmp = ((cos(k_m) * l) / k_m) * ((l * 2.0) / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.2e+160)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64(sin(k_m) * Float64(tan(k_m) * t)) / l))) / l));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(Float64(l * 2.0) / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.2e+160], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.19999999999999996e160

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 90.4

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

   10. Applied rewrites 90.4%

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

   if 8.19999999999999996e160 < k

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.0

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 74.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 82.9

          \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell \cdot 2}{\color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right)} \cdot k} \]

   8. Applied rewrites 82.9%

      \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\ell \cdot 2}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \left(\tan k\_m \cdot t\right)\\ \mathbf{if}\;k\_m \leq 2.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{t\_1}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \frac{t\_1 \cdot k\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (* (tan k_m) t))))
   (if (<= k_m 2.8e+116)
     (/ 2.0 (/ (* k_m (* k_m (/ t_1 l))) l))
     (/ 2.0 (/ (* k_m (/ (* t_1 k_m) l)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * (tan(k_m) * t);
	double tmp;
	if (k_m <= 2.8e+116) {
		tmp = 2.0 / ((k_m * (k_m * (t_1 / l))) / l);
	} else {
		tmp = 2.0 / ((k_m * ((t_1 * k_m) / l)) / l);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) * (tan(k_m) * t)
    if (k_m <= 2.8d+116) then
        tmp = 2.0d0 / ((k_m * (k_m * (t_1 / l))) / l)
    else
        tmp = 2.0d0 / ((k_m * ((t_1 * k_m) / l)) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * (Math.tan(k_m) * t);
	double tmp;
	if (k_m <= 2.8e+116) {
		tmp = 2.0 / ((k_m * (k_m * (t_1 / l))) / l);
	} else {
		tmp = 2.0 / ((k_m * ((t_1 * k_m) / l)) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.sin(k_m) * (math.tan(k_m) * t)
	tmp = 0
	if k_m <= 2.8e+116:
		tmp = 2.0 / ((k_m * (k_m * (t_1 / l))) / l)
	else:
		tmp = 2.0 / ((k_m * ((t_1 * k_m) / l)) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * Float64(tan(k_m) * t))
	tmp = 0.0
	if (k_m <= 2.8e+116)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(t_1 / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(Float64(t_1 * k_m) / l)) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) * (tan(k_m) * t);
	tmp = 0.0;
	if (k_m <= 2.8e+116)
		tmp = 2.0 / ((k_m * (k_m * (t_1 / l))) / l);
	else
		tmp = 2.0 / ((k_m * ((t_1 * k_m) / l)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2.8e+116], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(N[(t$95$1 * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.80000000000000004e116

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 90.4

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

   10. Applied rewrites 90.4%

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

   if 2.80000000000000004e116 < k

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 91.2

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

   10. Applied rewrites 91.2%

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

Alternative 3: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k\_m \cdot k\_m\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m \cdot k\_m}{\ell} \cdot \left(t \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 9.5e+113)
   (/ 2.0 (/ (* (* (* (tan k_m) k_m) (* (/ t l) (sin k_m))) k_m) l))
   (/ 2.0 (/ (* (/ (* k_m k_m) l) (* t (* (tan k_m) (sin k_m)))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.5e+113) {
		tmp = 2.0 / ((((tan(k_m) * k_m) * ((t / l) * sin(k_m))) * k_m) / l);
	} else {
		tmp = 2.0 / ((((k_m * k_m) / l) * (t * (tan(k_m) * sin(k_m)))) / l);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 9.5d+113) then
        tmp = 2.0d0 / ((((tan(k_m) * k_m) * ((t / l) * sin(k_m))) * k_m) / l)
    else
        tmp = 2.0d0 / ((((k_m * k_m) / l) * (t * (tan(k_m) * sin(k_m)))) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.5e+113) {
		tmp = 2.0 / ((((Math.tan(k_m) * k_m) * ((t / l) * Math.sin(k_m))) * k_m) / l);
	} else {
		tmp = 2.0 / ((((k_m * k_m) / l) * (t * (Math.tan(k_m) * Math.sin(k_m)))) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 9.5e+113:
		tmp = 2.0 / ((((math.tan(k_m) * k_m) * ((t / l) * math.sin(k_m))) * k_m) / l)
	else:
		tmp = 2.0 / ((((k_m * k_m) / l) * (t * (math.tan(k_m) * math.sin(k_m)))) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 9.5e+113)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) * k_m) * Float64(Float64(t / l) * sin(k_m))) * k_m) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / l) * Float64(t * Float64(tan(k_m) * sin(k_m)))) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 9.5e+113)
		tmp = 2.0 / ((((tan(k_m) * k_m) * ((t / l) * sin(k_m))) * k_m) / l);
	else
		tmp = 2.0 / ((((k_m * k_m) / l) * (t * (tan(k_m) * sin(k_m)))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 9.5e+113], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5000000000000001e113

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.3

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 87.3

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

   10. Applied rewrites 87.3%

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

   if 9.5000000000000001e113 < t

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. tan-quot N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

Alternative 4: 89.7% accurate, 1.4× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (/ (* k_m (* k_m (/ (* (sin k_m) (* (tan k_m) t)) l))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((k_m * (k_m * ((sin(k_m) * (tan(k_m) * t)) / l))) / l);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m * (k_m * ((sin(k_m) * (tan(k_m) * t)) / l))) / l)
end function

Java

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

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / ((k_m * (k_m * ((math.sin(k_m) * (math.tan(k_m) * t)) / l))) / l)

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 38.0%

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

4. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lift-pow.f64 N/A

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

   15. unpow2 N/A

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

   16. associate-*l/ N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. tan-quot N/A

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

   20. lift-tan.f64 N/A

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

8. Applied rewrites 87.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 90.4

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

10. Applied rewrites 90.4%

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

Alternative 5: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k\_m \cdot k\_m\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8e-119)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l))
   (/ 2.0 (/ (* (* (* (tan k_m) k_m) (* (/ t l) (sin k_m))) k_m) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e-119) {
		tmp = 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = 2.0 / ((((tan(k_m) * k_m) * ((t / l) * sin(k_m))) * k_m) / l);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8d-119) then
        tmp = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
    else
        tmp = 2.0d0 / ((((tan(k_m) * k_m) * ((t / l) * sin(k_m))) * k_m) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e-119) {
		tmp = 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = 2.0 / ((((Math.tan(k_m) * k_m) * ((t / l) * Math.sin(k_m))) * k_m) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8e-119:
		tmp = 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)
	else:
		tmp = 2.0 / ((((math.tan(k_m) * k_m) * ((t / l) * math.sin(k_m))) * k_m) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8e-119)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k_m) * k_m) * Float64(Float64(t / l) * sin(k_m))) * k_m) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8e-119)
		tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
	else
		tmp = 2.0 / ((((tan(k_m) * k_m) * ((t / l) * sin(k_m))) * k_m) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8e-119], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000001e-119

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.9

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

   11. Applied rewrites 72.9%

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

   if 8.0000000000000001e-119 < k

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.3

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 87.3

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

   10. Applied rewrites 87.3%

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

Alternative 6: 88.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8e-119)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l))
   (/ 2.0 (/ (* k_m (* k_m (* (* (tan k_m) (sin k_m)) (/ t l)))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e-119) {
		tmp = 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = 2.0 / ((k_m * (k_m * ((tan(k_m) * sin(k_m)) * (t / l)))) / l);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8d-119) then
        tmp = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
    else
        tmp = 2.0d0 / ((k_m * (k_m * ((tan(k_m) * sin(k_m)) * (t / l)))) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e-119) {
		tmp = 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = 2.0 / ((k_m * (k_m * ((Math.tan(k_m) * Math.sin(k_m)) * (t / l)))) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8e-119:
		tmp = 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)
	else:
		tmp = 2.0 / ((k_m * (k_m * ((math.tan(k_m) * math.sin(k_m)) * (t / l)))) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8e-119)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64(tan(k_m) * sin(k_m)) * Float64(t / l)))) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8e-119)
		tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
	else
		tmp = 2.0 / ((k_m * (k_m * ((tan(k_m) * sin(k_m)) * (t / l)))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8e-119], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000001e-119

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.9

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

   11. Applied rewrites 72.9%

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

   if 8.0000000000000001e-119 < k

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

Alternative 7: 86.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \frac{t}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8e-119)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l))
   (* (/ 2.0 (* (* (* (tan k_m) (sin k_m)) (/ t l)) (* k_m k_m))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e-119) {
		tmp = 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = (2.0 / (((tan(k_m) * sin(k_m)) * (t / l)) * (k_m * k_m))) * l;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8d-119) then
        tmp = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
    else
        tmp = (2.0d0 / (((tan(k_m) * sin(k_m)) * (t / l)) * (k_m * k_m))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8e-119) {
		tmp = 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = (2.0 / (((Math.tan(k_m) * Math.sin(k_m)) * (t / l)) * (k_m * k_m))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8e-119:
		tmp = 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)
	else:
		tmp = (2.0 / (((math.tan(k_m) * math.sin(k_m)) * (t / l)) * (k_m * k_m))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8e-119)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(t / l)) * Float64(k_m * k_m))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8e-119)
		tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
	else
		tmp = (2.0 / (((tan(k_m) * sin(k_m)) * (t / l)) * (k_m * k_m))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8e-119], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000001e-119

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.9

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

   11. Applied rewrites 72.9%

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

   if 8.0000000000000001e-119 < k

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 83.3%

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

Alternative 8: 85.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\frac{t}{\ell} \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.4e-93)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l))
   (/ (+ l l) (* (* k_m k_m) (* (/ t l) (* (sin k_m) (tan k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e-93) {
		tmp = 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = (l + l) / ((k_m * k_m) * ((t / l) * (sin(k_m) * tan(k_m))));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.4d-93) then
        tmp = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
    else
        tmp = (l + l) / ((k_m * k_m) * ((t / l) * (sin(k_m) * tan(k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e-93) {
		tmp = 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = (l + l) / ((k_m * k_m) * ((t / l) * (Math.sin(k_m) * Math.tan(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.4e-93:
		tmp = 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)
	else:
		tmp = (l + l) / ((k_m * k_m) * ((t / l) * (math.sin(k_m) * math.tan(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.4e-93)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l));
	else
		tmp = Float64(Float64(l + l) / Float64(Float64(k_m * k_m) * Float64(Float64(t / l) * Float64(sin(k_m) * tan(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.4e-93)
		tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
	else
		tmp = (l + l) / ((k_m * k_m) * ((t / l) * (sin(k_m) * tan(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.4e-93], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l + l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.40000000000000001e-93

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.9

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

   11. Applied rewrites 72.9%

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

   if 3.40000000000000001e-93 < k

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

   10. Applied rewrites 83.5%

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

Alternative 9: 73.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.32 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1.32e+100)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l))
   (/ (* (* (* (cos k_m) l) l) 2.0) (* (* (* (- 0.5 0.5) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.32e+100) {
		tmp = 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 1.32d+100) then
        tmp = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
    else
        tmp = (((cos(k_m) * l) * l) * 2.0d0) / ((((0.5d0 - 0.5d0) * t) * k_m) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.32e+100) {
		tmp = 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = (((Math.cos(k_m) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1.32e+100:
		tmp = 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)
	else:
		tmp = (((math.cos(k_m) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 1.32e+100)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1.32e+100)
		tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
	else
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1.32e+100], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.32e100

   1. Initial program 36.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l/ N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

   8. Applied rewrites 87.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 72.9

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

   11. Applied rewrites 72.9%

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

   if 1.32e100 < l

   1. Initial program 36.0%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 74.0

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      15. lower-*.f64 74.0

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(1 + -0.5 \cdot {k\_m}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 2.4e+194)
   (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l))
   (/
    (* (* (* (+ 1.0 (* -0.5 (pow k_m 2.0))) l) l) 2.0)
    (* (* (* (- 0.5 0.5) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 2.4e+194) {
		tmp = 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = ((((1.0 + (-0.5 * pow(k_m, 2.0))) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 2.4d+194) then
        tmp = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
    else
        tmp = ((((1.0d0 + ((-0.5d0) * (k_m ** 2.0d0))) * l) * l) * 2.0d0) / ((((0.5d0 - 0.5d0) * t) * k_m) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 2.4e+194) {
		tmp = 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
	} else {
		tmp = ((((1.0 + (-0.5 * Math.pow(k_m, 2.0))) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 2.4e+194:
		tmp = 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)
	else:
		tmp = ((((1.0 + (-0.5 * math.pow(k_m, 2.0))) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 2.4e+194)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + Float64(-0.5 * (k_m ^ 2.0))) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 2.4e+194)
		tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
	else
		tmp = ((((1.0 + (-0.5 * (k_m ^ 2.0))) * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 2.4e+194], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + N[(-0.5 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4e194

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

   3. Applied rewrites 38.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(k \cdot \frac{k}{t \cdot t}\right)\right) \cdot \left(\sin k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}}} \]

   4. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell \cdot \cos k}}}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell} \cdot \cos k}}{\ell}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \color{blue}{\cos k}}}{\ell}} \]

      8. lower-cos.f64 82.3

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   6. Applied rewrites 82.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell \cdot \cos k}}}{\ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell} \cdot \cos k}}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}}}{\ell}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}}\right)}{\ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \cos k}\right)}{\ell}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\ell} \cdot \cos k}\right)}{\ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\ell \cdot \color{blue}{\cos k}}\right)}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{\ell}}\right)}{\ell}} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell}}\right)\right)}{\ell}} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

      16. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{\color{blue}{t}}{\ell}\right)\right)}{\ell}} \]

      17. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

      18. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

      19. tan-quot N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

      20. lift-tan.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   8. Applied rewrites 87.3%

      \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}}{\ell}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\color{blue}{\ell}}\right)}{\ell}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}{\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}{\ell}} \]

       3. lower-pow.f64 72.9

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}{\ell}} \]

   11. Applied rewrites 72.9%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\color{blue}{\ell}}\right)}{\ell}} \]

   if 2.4e194 < l

   1. Initial program 36.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

      10. lower-sin.f64 74.0

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. Applied rewrites 74.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 74.0

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      15. lower-*.f64 74.0

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{\left(\left(\left(1 + \frac{-1}{2} \cdot {k}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\left(1 + \frac{-1}{2} \cdot {k}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(1 + \frac{-1}{2} \cdot {k}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]

       3. lower-pow.f64 33.4

          \[\leadsto \frac{\left(\left(\left(1 + -0.5 \cdot {k}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \]

   12. Applied rewrites 33.4%

       \[\leadsto \frac{\left(\left(\left(1 + -0.5 \cdot {k}^{2}\right) \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\color{blue}{0.5} - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{k\_m \cdot \left(k\_m \cdot \frac{{k\_m}^{2} \cdot t}{\ell}\right)}{\ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (/ (* k_m (* k_m (/ (* (pow k_m 2.0) t) l))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((k_m * (k_m * ((pow(k_m, 2.0) * t) / l))) / l);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m * (k_m * (((k_m ** 2.0d0) * t) / l))) / l)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / ((k_m * (k_m * ((Math.pow(k_m, 2.0) * t) / l))) / l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / ((k_m * (k_m * ((math.pow(k_m, 2.0) * t) / l))) / l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64((k_m ^ 2.0) * t) / l))) / l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((k_m * (k_m * (((k_m ^ 2.0) * t) / l))) / l);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right)} \]

   7. associate-*l/ N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

3. Applied rewrites 38.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(k \cdot \frac{k}{t \cdot t}\right)\right) \cdot \left(\sin k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}}} \]

4. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell \cdot \cos k}}}{\ell}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell} \cdot \cos k}}{\ell}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \color{blue}{\cos k}}}{\ell}} \]

   8. lower-cos.f64 82.3

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

6. Applied rewrites 82.3%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell \cdot \cos k}}}{\ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell} \cdot \cos k}}{\ell}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}}}{\ell}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}}\right)}{\ell}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \cos k}\right)}{\ell}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\ell} \cdot \cos k}\right)}{\ell}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\ell \cdot \color{blue}{\cos k}}\right)}{\ell}} \]

   12. *-commutative N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{\ell}}\right)}{\ell}} \]

   13. times-frac N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell}}\right)\right)}{\ell}} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   15. unpow2 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   16. associate-*l/ N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{\color{blue}{t}}{\ell}\right)\right)}{\ell}} \]

   17. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   18. lift-cos.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   19. tan-quot N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   20. lift-tan.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

8. Applied rewrites 87.3%

   \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}}{\ell}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\color{blue}{\ell}}\right)}{\ell}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}{\ell}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}{\ell}} \]

    3. lower-pow.f64 72.9

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\ell}\right)}{\ell}} \]

11. Applied rewrites 72.9%

    \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{k}^{2} \cdot t}{\color{blue}{\ell}}\right)}{\ell}} \]
12. Add Preprocessing

Alternative 12: 70.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{k\_m \cdot \frac{{k\_m}^{3} \cdot t}{\ell}}{\ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (/ (* k_m (/ (* (pow k_m 3.0) t) l)) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((k_m * ((pow(k_m, 3.0) * t) / l)) / l);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m * (((k_m ** 3.0d0) * t) / l)) / l)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / ((k_m * ((Math.pow(k_m, 3.0) * t) / l)) / l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / ((k_m * ((math.pow(k_m, 3.0) * t) / l)) / l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(k_m * Float64(Float64((k_m ^ 3.0) * t) / l)) / l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((k_m * (((k_m ^ 3.0) * t) / l)) / l);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(k$95$m * N[(N[(N[Power[k$95$m, 3.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right)} \]

   7. associate-*l/ N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \frac{{t}^{3} \cdot \sin k}{\ell}}{\ell}}} \]

3. Applied rewrites 38.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(k \cdot \frac{k}{t \cdot t}\right)\right) \cdot \left(\sin k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}}} \]

4. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell \cdot \cos k}}}{\ell}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell} \cdot \cos k}}{\ell}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \color{blue}{\cos k}}}{\ell}} \]

   8. lower-cos.f64 82.3

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}} \]

6. Applied rewrites 82.3%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell \cdot \cos k}}}{\ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\ell} \cdot \cos k}}{\ell}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}}}{\ell}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\ell \cdot \cos k}}\right)}{\ell}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell} \cdot \cos k}\right)}{\ell}} \]

   10. *-commutative N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\ell} \cdot \cos k}\right)}{\ell}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\ell \cdot \color{blue}{\cos k}}\right)}{\ell}} \]

   12. *-commutative N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{\ell}}\right)}{\ell}} \]

   13. times-frac N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell}}\right)\right)}{\ell}} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   15. unpow2 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\frac{\sin k \cdot \sin k}{\cos k} \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   16. associate-*l/ N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{\color{blue}{t}}{\ell}\right)\right)}{\ell}} \]

   17. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   18. lift-cos.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\frac{\sin k}{\cos k} \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   19. tan-quot N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

   20. lift-tan.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}{\ell}} \]

8. Applied rewrites 87.3%

   \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell}\right)\right)}}{\ell}} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{k \cdot \frac{{k}^{3} \cdot t}{\color{blue}{\ell}}}{\ell}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \frac{{k}^{3} \cdot t}{\ell}}{\ell}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{k \cdot \frac{{k}^{3} \cdot t}{\ell}}{\ell}} \]

    3. lower-pow.f64 70.5

       \[\leadsto \frac{2}{\frac{k \cdot \frac{{k}^{3} \cdot t}{\ell}}{\ell}} \]

11. Applied rewrites 70.5%

    \[\leadsto \frac{2}{\frac{k \cdot \frac{{k}^{3} \cdot t}{\color{blue}{\ell}}}{\ell}} \]
12. Add Preprocessing

Alternative 13: 68.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{{k\_m}^{4} \cdot t} \cdot \left(\ell + \ell\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (/ l (* (pow k_m 4.0) t)) (+ l l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (pow(k_m, 4.0) * t)) * (l + l);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l / ((k_m ** 4.0d0) * t)) * (l + l)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / (Math.pow(k_m, 4.0) * t)) * (l + l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (math.pow(k_m, 4.0) * t)) * (l + l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64((k_m ^ 4.0) * t)) * Float64(l + l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / ((k_m ^ 4.0) * t)) * (l + l);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot \color{blue}{t}} \]

   5. lower-pow.f64 62.8

      \[\leadsto 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

4. Applied rewrites 62.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   3. lower-*.f64 62.8

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   9. lower-/.f64 68.7

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

6. Applied rewrites 68.7%

   \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \color{blue}{2} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot 2 \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right) \cdot 2 \]

   3. lower-*.f64 68.7

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right) \cdot 2 \]

8. Applied rewrites 68.7%

   \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right) \cdot 2 \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right) \cdot \color{blue}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right) \cdot 2 \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \left(1 + \color{blue}{1}\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(1 \cdot \ell + \color{blue}{1 \cdot \ell}\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\frac{1}{1} \cdot \ell + 1 \cdot \ell\right) \]

   8. associate-/r/ N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\frac{1}{\frac{1}{\ell}} + \color{blue}{1} \cdot \ell\right) \]

   9. remove-double-div N/A

      \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \color{blue}{1} \cdot \ell\right) \]

   10. metadata-eval N/A

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \frac{1}{1} \cdot \ell\right) \]

   11. associate-/r/ N/A

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \frac{1}{\color{blue}{\frac{1}{\ell}}}\right) \]

   12. remove-double-div N/A

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right) \]

   13. lower-+.f64 68.7

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \color{blue}{\ell}\right) \]

10. Applied rewrites 68.7%

    \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(\ell + \ell\right)} \]
11. Add Preprocessing

Alternative 14: 33.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\left(1 \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* (* 1.0 l) l) 2.0) (* (* (* (- 0.5 0.5) t) k_m) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((1.0 * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (((1.0d0 * l) * l) * 2.0d0) / ((((0.5d0 - 0.5d0) * t) * k_m) * k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((1.0 * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((1.0 * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(1.0 * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((1.0 * l) * l) * 2.0) / ((((0.5 - 0.5) * t) * k_m) * k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(1.0 * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   7. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

   9. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

   10. lower-sin.f64 74.0

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

4. Applied rewrites 74.0%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-*.f64 74.0

      \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{\left({\ell}^{2} \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   9. pow2 N/A

      \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   15. lower-*.f64 74.0

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   17. *-commutative N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   18. lift-pow.f64 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{\color{blue}{2}}} \]

   19. unpow2 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(t \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   20. associate-*r* N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right) \cdot \color{blue}{k}} \]

6. Applied rewrites 69.8%

   \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 35.5%

   \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \]

10. Taylor expanded in k around 0

    \[\leadsto \frac{\left(\left(1 \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot t\right) \cdot k\right) \cdot k} \]
11. Step-by-step derivation

12. Applied rewrites 33.3%

    \[\leadsto \frac{\left(\left(1 \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(\color{blue}{0.5} - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))