Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 81.3%

Time: 6.8s

Alternatives: 14

Speedup: 6.6×

• 27.8% 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: 55.0% 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: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \log l\_m \cdot -2\\ t_3 := \log t\_m \cdot 3\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.85 \cdot 10^{-97}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 4.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\frac{\mathsf{fma}\left({\log t\_m}^{3}, 27, {t\_2}^{3}\right)}{\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2 - t\_3 \cdot t\_2\right)}} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot l\_m} \cdot \frac{k\_m}{l\_m}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (log l_m) -2.0)) (t_3 (* (log t_m) 3.0)))
   (*
    t_s
    (if (<= k_m 2.85e-97)
      (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))
      (if (<= k_m 4.2e+28)
        (/
         2.0
         (*
          (*
           (*
            (exp
             (/
              (fma (pow (log t_m) 3.0) 27.0 (pow t_2 3.0))
              (fma t_3 t_3 (- (* t_2 t_2) (* t_3 t_2)))))
            (sin k_m))
           (tan k_m))
          (+ (fma (/ k_m t_m) (/ k_m t_m) 1.0) 1.0)))
        (/
         2.0
         (*
          (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t_m)
          (* (/ k_m (* (cos k_m) l_m)) (/ k_m l_m)))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = log(l_m) * -2.0;
	double t_3 = log(t_m) * 3.0;
	double tmp;
	if (k_m <= 2.85e-97) {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	} else if (k_m <= 4.2e+28) {
		tmp = 2.0 / (((exp((fma(pow(log(t_m), 3.0), 27.0, pow(t_2, 3.0)) / fma(t_3, t_3, ((t_2 * t_2) - (t_3 * t_2))))) * sin(k_m)) * tan(k_m)) * (fma((k_m / t_m), (k_m / t_m), 1.0) + 1.0));
	} else {
		tmp = 2.0 / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t_m) * ((k_m / (cos(k_m) * l_m)) * (k_m / l_m)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(log(l_m) * -2.0)
	t_3 = Float64(log(t_m) * 3.0)
	tmp = 0.0
	if (k_m <= 2.85e-97)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	elseif (k_m <= 4.2e+28)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(fma((log(t_m) ^ 3.0), 27.0, (t_2 ^ 3.0)) / fma(t_3, t_3, Float64(Float64(t_2 * t_2) - Float64(t_3 * t_2))))) * sin(k_m)) * tan(k_m)) * Float64(fma(Float64(k_m / t_m), Float64(k_m / t_m), 1.0) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t_m) * Float64(Float64(k_m / Float64(cos(k_m) * l_m)) * Float64(k_m / l_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[Log[l$95$m], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.85e-97], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.2e+28], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Power[N[Log[t$95$m], $MachinePrecision], 3.0], $MachinePrecision] * 27.0 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3 + N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.85e-97

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if 2.85e-97 < k < 4.19999999999999978e28

   1. Initial program 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.8

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

   3. Applied rewrites 70.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 70.9

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

   5. Applied rewrites 70.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\left(e^{\frac{\mathsf{fma}\left({\log t}^{3}, \color{blue}{27}, {\left(-2 \cdot \log \ell\right)}^{3}\right)}{\left(\log t \cdot 3\right) \cdot \left(\log t \cdot 3\right) + \left(\left(-2 \cdot \log \ell\right) \cdot \left(-2 \cdot \log \ell\right) - \left(\log t \cdot 3\right) \cdot \left(-2 \cdot \log \ell\right)\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\frac{\mathsf{fma}\left({\log t}^{3}, 27, \color{blue}{{\left(-2 \cdot \log \ell\right)}^{3}}\right)}{\left(\log t \cdot 3\right) \cdot \left(\log t \cdot 3\right) + \left(\left(-2 \cdot \log \ell\right) \cdot \left(-2 \cdot \log \ell\right) - \left(\log t \cdot 3\right) \cdot \left(-2 \cdot \log \ell\right)\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(e^{\frac{\mathsf{fma}\left({\log t}^{3}, 27, {\color{blue}{\left(\log \ell \cdot -2\right)}}^{3}\right)}{\left(\log t \cdot 3\right) \cdot \left(\log t \cdot 3\right) + \left(\left(-2 \cdot \log \ell\right) \cdot \left(-2 \cdot \log \ell\right) - \left(\log t \cdot 3\right) \cdot \left(-2 \cdot \log \ell\right)\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\frac{\mathsf{fma}\left({\log t}^{3}, 27, {\color{blue}{\left(\log \ell \cdot -2\right)}}^{3}\right)}{\left(\log t \cdot 3\right) \cdot \left(\log t \cdot 3\right) + \left(\left(-2 \cdot \log \ell\right) \cdot \left(-2 \cdot \log \ell\right) - \left(\log t \cdot 3\right) \cdot \left(-2 \cdot \log \ell\right)\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\frac{\mathsf{fma}\left({\log t}^{3}, 27, {\left(\color{blue}{\log \ell} \cdot -2\right)}^{3}\right)}{\left(\log t \cdot 3\right) \cdot \left(\log t \cdot 3\right) + \left(\left(-2 \cdot \log \ell\right) \cdot \left(-2 \cdot \log \ell\right) - \left(\log t \cdot 3\right) \cdot \left(-2 \cdot \log \ell\right)\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]

   9. Applied rewrites 70.8%

      \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\frac{\mathsf{fma}\left({\log t}^{3}, 27, {\left(\log \ell \cdot -2\right)}^{3}\right)}{\mathsf{fma}\left(\log t \cdot 3, \log t \cdot 3, \left(\log \ell \cdot -2\right) \cdot \left(\log \ell \cdot -2\right) - \left(\log t \cdot 3\right) \cdot \left(\log \ell \cdot -2\right)\right)}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]

   if 4.19999999999999978e28 < k

   1. Initial program 55.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 66.6

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

   8. Applied rewrites 66.6%

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

Alternative 2: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-103}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 4.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot l\_m} \cdot \frac{k\_m}{l\_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 6.2e-103)
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))
    (if (<= k_m 4.2e+28)
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (tan k_m))
        (fma (/ k_m t_m) (/ k_m t_m) 2.0)))
      (/
       2.0
       (*
        (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t_m)
        (* (/ k_m (* (cos k_m) l_m)) (/ k_m l_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 6.2e-103) {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	} else if (k_m <= 4.2e+28) {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * fma((k_m / t_m), (k_m / t_m), 2.0));
	} else {
		tmp = 2.0 / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t_m) * ((k_m / (cos(k_m) * l_m)) * (k_m / l_m)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 6.2e-103)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	elseif (k_m <= 4.2e+28)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * fma(Float64(k_m / t_m), Float64(k_m / t_m), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t_m) * Float64(Float64(k_m / Float64(cos(k_m) * l_m)) * Float64(k_m / l_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 6.2e-103], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.2e+28], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.2000000000000003e-103

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if 6.2000000000000003e-103 < k < 4.19999999999999978e28

   1. Initial program 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.8

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

   3. Applied rewrites 70.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 70.9

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

   5. Applied rewrites 70.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 70.9

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

   9. Applied rewrites 70.9%

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

   if 4.19999999999999978e28 < k

   1. Initial program 55.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 66.6

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

   8. Applied rewrites 66.6%

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

Alternative 3: 80.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 2500000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m, t\_m\right), k\_m \cdot k\_m, \left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right)}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(\frac{k\_m}{\cos k\_m \cdot l\_m} \cdot \frac{k\_m}{l\_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.5e-67)
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))
    (if (<= k_m 2500000000000.0)
      (/
       2.0
       (*
        (/
         (/
          (fma
           (fma (* (* t_m t_m) 0.3333333333333333) t_m t_m)
           (* k_m k_m)
           (* (* (* t_m t_m) 2.0) t_m))
          l_m)
         l_m)
        (* k_m k_m)))
      (/
       2.0
       (*
        (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t_m)
        (* (/ k_m (* (cos k_m) l_m)) (/ k_m l_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.5e-67) {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	} else if (k_m <= 2500000000000.0) {
		tmp = 2.0 / (((fma(fma(((t_m * t_m) * 0.3333333333333333), t_m, t_m), (k_m * k_m), (((t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * (k_m * k_m));
	} else {
		tmp = 2.0 / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t_m) * ((k_m / (cos(k_m) * l_m)) * (k_m / l_m)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-67)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	elseif (k_m <= 2500000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m, t_m), Float64(k_m * k_m), Float64(Float64(Float64(t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t_m) * Float64(Float64(k_m / Float64(cos(k_m) * l_m)) * Float64(k_m / l_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.5e-67], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2500000000000.0], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000016e-67

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if 1.50000000000000016e-67 < k < 2.5e12

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 62.5%

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

   if 2.5e12 < k

   1. Initial program 55.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 66.6

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

   8. Applied rewrites 66.6%

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

Alternative 4: 76.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 2500000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m, t\_m\right), k\_m \cdot k\_m, \left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right)}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(k\_m \cdot \frac{k\_m}{\left(\cos k\_m \cdot l\_m\right) \cdot l\_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.5e-67)
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))
    (if (<= k_m 2500000000000.0)
      (/
       2.0
       (*
        (/
         (/
          (fma
           (fma (* (* t_m t_m) 0.3333333333333333) t_m t_m)
           (* k_m k_m)
           (* (* (* t_m t_m) 2.0) t_m))
          l_m)
         l_m)
        (* k_m k_m)))
      (/
       2.0
       (*
        (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t_m)
        (* k_m (/ k_m (* (* (cos k_m) l_m) l_m)))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.5e-67) {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	} else if (k_m <= 2500000000000.0) {
		tmp = 2.0 / (((fma(fma(((t_m * t_m) * 0.3333333333333333), t_m, t_m), (k_m * k_m), (((t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * (k_m * k_m));
	} else {
		tmp = 2.0 / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t_m) * (k_m * (k_m / ((cos(k_m) * l_m) * l_m))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-67)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	elseif (k_m <= 2500000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m, t_m), Float64(k_m * k_m), Float64(Float64(Float64(t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t_m) * Float64(k_m * Float64(k_m / Float64(Float64(cos(k_m) * l_m) * l_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.5e-67], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2500000000000.0], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000016e-67

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if 1.50000000000000016e-67 < k < 2.5e12

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 62.5%

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

   if 2.5e12 < k

   1. Initial program 55.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

      15. lift-cos.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 58.3

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

   8. Applied rewrites 58.3%

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

Alternative 5: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k\_m \leq 205000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m, t\_m\right), k\_m \cdot k\_m, \left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right)}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot l\_m\right) \cdot l\_m\right)}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.5e-67)
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))
    (if (<= k_m 205000000000.0)
      (/
       2.0
       (*
        (/
         (/
          (fma
           (fma (* (* t_m t_m) 0.3333333333333333) t_m t_m)
           (* k_m k_m)
           (* (* (* t_m t_m) 2.0) t_m))
          l_m)
         l_m)
        (* k_m k_m)))
      (/
       (* 2.0 (* (* (cos k_m) l_m) l_m))
       (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t_m) k_m) k_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.5e-67) {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	} else if (k_m <= 205000000000.0) {
		tmp = 2.0 / (((fma(fma(((t_m * t_m) * 0.3333333333333333), t_m, t_m), (k_m * k_m), (((t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * (k_m * k_m));
	} else {
		tmp = (2.0 * ((cos(k_m) * l_m) * l_m)) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t_m) * k_m) * k_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-67)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	elseif (k_m <= 205000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m, t_m), Float64(k_m * k_m), Float64(Float64(Float64(t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l_m) * l_m)) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t_m) * k_m) * k_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.5e-67], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 205000000000.0], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.50000000000000016e-67

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if 1.50000000000000016e-67 < k < 2.05e11

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 62.5%

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

   if 2.05e11 < k

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. sqr-sin-a-rev N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

   7. Applied rewrites 59.3%

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

Alternative 6: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m, t\_m\right), k\_m \cdot k\_m, \left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right)}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+148}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot k\_m\right) \cdot \tan k\_m\right) \cdot \left(\mathsf{fma}\left(\frac{k\_m}{t\_m}, \frac{k\_m}{t\_m}, 1\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.6e-29)
    (/
     2.0
     (*
      (/
       (/
        (fma
         (fma (* (* t_m t_m) 0.3333333333333333) t_m t_m)
         (* k_m k_m)
         (* (* (* t_m t_m) 2.0) t_m))
        l_m)
       l_m)
      (* k_m k_m)))
    (if (<= t_m 4e+148)
      (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))
      (/
       2.0
       (*
        (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) k_m) (tan k_m))
        (+ (fma (/ k_m t_m) (/ k_m t_m) 1.0) 1.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 3.6e-29) {
		tmp = 2.0 / (((fma(fma(((t_m * t_m) * 0.3333333333333333), t_m, t_m), (k_m * k_m), (((t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * (k_m * k_m));
	} else if (t_m <= 4e+148) {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * k_m) * tan(k_m)) * (fma((k_m / t_m), (k_m / t_m), 1.0) + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 3.6e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m, t_m), Float64(k_m * k_m), Float64(Float64(Float64(t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * Float64(k_m * k_m)));
	elseif (t_m <= 4e+148)
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * k_m) * tan(k_m)) * Float64(fma(Float64(k_m / t_m), Float64(k_m / t_m), 1.0) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-29], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+148], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.59999999999999974e-29

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 62.5%

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

   if 3.59999999999999974e-29 < t < 4.0000000000000002e148

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

   if 4.0000000000000002e148 < t

   1. Initial program 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.8

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

   3. Applied rewrites 70.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 70.9

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

   5. Applied rewrites 70.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 64.4%

       \[\leadsto \frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \color{blue}{k}\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right) + 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m, t\_m\right), k\_m \cdot k\_m, \left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right)}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.6e-29)
    (/
     2.0
     (*
      (/
       (/
        (fma
         (fma (* (* t_m t_m) 0.3333333333333333) t_m t_m)
         (* k_m k_m)
         (* (* (* t_m t_m) 2.0) t_m))
        l_m)
       l_m)
      (* k_m k_m)))
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 3.6e-29) {
		tmp = 2.0 / (((fma(fma(((t_m * t_m) * 0.3333333333333333), t_m, t_m), (k_m * k_m), (((t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * (k_m * k_m));
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 3.6e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m, t_m), Float64(k_m * k_m), Float64(Float64(Float64(t_m * t_m) * 2.0) * t_m)) / l_m) / l_m) * Float64(k_m * k_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-29], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.59999999999999974e-29

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 62.5%

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

   if 3.59999999999999974e-29 < t

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. unpow3 N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 62.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.5

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

   8. Applied rewrites 62.5%

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

Alternative 8: 68.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right) \cdot \left(k\_m \cdot k\_m\right)}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-41)
    (/
     2.0
     (*
      (/
       (/ (* (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m) (* k_m k_m)) l_m)
       l_m)
      (* k_m k_m)))
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.85e-41) {
		tmp = 2.0 / (((((fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m) * (k_m * k_m)) / l_m) / l_m) * (k_m * k_m));
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.85e-41)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m) * Float64(k_m * k_m)) / l_m) / l_m) * Float64(k_m * k_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-41], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-41

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow3 N/A

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

      4. lower-*.f64 N/A

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

      5. pow3 N/A

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

      6. unpow3 N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{1}{3} \cdot {t}^{2}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({t}^{2} \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({t}^{2} \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{3}, t, t\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      15. lift-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

   9. Applied rewrites 60.1%

      \[\leadsto \frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\ell}}{\ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   if 1.8500000000000001e-41 < t

   1. Initial program 55.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      12. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      13. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 55.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. Applied rewrites 55.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      5. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      6. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left({t}^{2} \cdot t\right) \cdot k\right)} \]

      10. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      13. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      15. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      18. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   8. Applied rewrites 62.5%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.3333333333333333, 1\right) \cdot t\_m\right) \cdot \left(k\_m \cdot k\_m\right)}{l\_m \cdot l\_m} \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-41)
    (/
     2.0
     (*
      (*
       (/
        (* (* (fma (* t_m t_m) 0.3333333333333333 1.0) t_m) (* k_m k_m))
        (* l_m l_m))
       k_m)
      k_m))
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.85e-41) {
		tmp = 2.0 / (((((fma((t_m * t_m), 0.3333333333333333, 1.0) * t_m) * (k_m * k_m)) / (l_m * l_m)) * k_m) * k_m);
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.85e-41)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 0.3333333333333333, 1.0) * t_m) * Float64(k_m * k_m)) / Float64(l_m * l_m)) * k_m) * k_m));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-41], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-41

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot t, t\right), k \cdot k, 2 \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t + \frac{1}{3} \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(t + \frac{1}{3} \cdot {t}^{3}\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\frac{1}{3} \cdot {t}^{3} + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      3. pow3 N/A

         \[\leadsto \frac{2}{\frac{\left(\frac{1}{3} \cdot \left(\left(t \cdot t\right) \cdot t\right) + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\frac{1}{3} \cdot \left(\left(t \cdot t\right) \cdot t\right) + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      5. pow3 N/A

         \[\leadsto \frac{2}{\frac{\left(\frac{1}{3} \cdot {t}^{3} + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      6. unpow3 N/A

         \[\leadsto \frac{2}{\frac{\left(\frac{1}{3} \cdot \left(\left(t \cdot t\right) \cdot t\right) + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\frac{1}{3} \cdot \left({t}^{2} \cdot t\right) + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{3} \cdot {t}^{2}\right) \cdot t + t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{1}{3} \cdot {t}^{2}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({t}^{2} \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({t}^{2} \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{3}, t, t\right) \cdot {k}^{2}}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{3}, t, t\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      15. lift-*.f64 55.5

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.3333333333333333, t, t\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   7. Applied rewrites 55.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.3333333333333333, t, t\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 55.5%

      \[\leadsto \frac{2}{\left(\frac{\left(\mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot k\right) \cdot \color{blue}{k}} \]

   if 1.8500000000000001e-41 < t

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lower-*.f64 51.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      10. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      12. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      13. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 55.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. Applied rewrites 55.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      5. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      6. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left({t}^{2} \cdot t\right) \cdot k\right)} \]

      10. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      13. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      15. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      18. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   8. Applied rewrites 62.5%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 65.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-41)
    (/ 2.0 (* (/ (* (* k_m k_m) t_m) (* l_m l_m)) (* k_m k_m)))
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.85e-41) {
		tmp = 2.0 / ((((k_m * k_m) * t_m) / (l_m * l_m)) * (k_m * k_m));
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.85d-41) then
        tmp = 2.0d0 / ((((k_m * k_m) * t_m) / (l_m * l_m)) * (k_m * k_m))
    else
        tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.85e-41) {
		tmp = 2.0 / ((((k_m * k_m) * t_m) / (l_m * l_m)) * (k_m * k_m));
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 1.85e-41:
		tmp = 2.0 / ((((k_m * k_m) * t_m) / (l_m * l_m)) * (k_m * k_m))
	else:
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.85e-41)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) / Float64(l_m * l_m)) * Float64(k_m * k_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 1.85e-41)
		tmp = 2.0 / ((((k_m * k_m) * t_m) / (l_m * l_m)) * (k_m * k_m));
	else
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-41], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-41

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot t, t\right), k \cdot k, 2 \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 54.0

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   7. Applied rewrites 54.0%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   if 1.8500000000000001e-41 < t

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lower-*.f64 51.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      10. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      12. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      13. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 55.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. Applied rewrites 55.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      5. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      6. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left({t}^{2} \cdot t\right) \cdot k\right)} \]

      10. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      13. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      15. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      18. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   8. Applied rewrites 62.5%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-41)
    (/ 2.0 (* (* (* k_m k_m) (* k_m k_m)) (/ t_m (* l_m l_m))))
    (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.85e-41) {
		tmp = 2.0 / (((k_m * k_m) * (k_m * k_m)) * (t_m / (l_m * l_m)));
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.85d-41) then
        tmp = 2.0d0 / (((k_m * k_m) * (k_m * k_m)) * (t_m / (l_m * l_m)))
    else
        tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.85e-41) {
		tmp = 2.0 / (((k_m * k_m) * (k_m * k_m)) * (t_m / (l_m * l_m)));
	} else {
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 1.85e-41:
		tmp = 2.0 / (((k_m * k_m) * (k_m * k_m)) * (t_m / (l_m * l_m)))
	else:
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.85e-41)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(k_m * k_m)) * Float64(t_m / Float64(l_m * l_m))));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 1.85e-41)
		tmp = 2.0 / (((k_m * k_m) * (k_m * k_m)) * (t_m / (l_m * l_m)));
	else
		tmp = l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-41], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-41

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot \sin k\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. sqr-sin-a N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot {k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot {\color{blue}{\ell}}^{2}}} \]

      17. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}} \]

      18. lift-*.f64 57.0

          \[\leadsto \frac{2}{\frac{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \color{blue}{\ell}\right)}} \]

   4. Applied rewrites 57.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\color{blue}{{\ell}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\color{blue}{{\ell}^{2}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2}{{k}^{\left(2 + 2\right)} \cdot \frac{t}{{\ell}^{2}}} \]

      4. pow-prod-up N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot {k}^{2}\right) \cdot \frac{t}{{\color{blue}{\ell}}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot {k}^{2}\right) \cdot \frac{t}{{\color{blue}{\ell}}^{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot \frac{t}{{\ell}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot \frac{t}{{\ell}^{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{{\ell}^{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{{\ell}^{\color{blue}{2}}}} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}} \]

      12. lift-*.f64 51.9

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}} \]

   7. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}} \]

   if 1.8500000000000001e-41 < t

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lower-*.f64 51.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      10. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      12. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      13. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 55.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. Applied rewrites 55.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      5. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      6. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

      8. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left({t}^{2} \cdot t\right) \cdot k\right)} \]

      10. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

      13. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

      15. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      18. lower-*.f64 62.5

          \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   8. Applied rewrites 62.5%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* l_m (/ l_m (* k_m (* (* t_m t_m) (* k_m t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * (l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m)))));
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * (l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m)))))
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * (l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m)))));
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * (l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m)))))

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(k_m * Float64(Float64(t_m * t_m) * Float64(k_m * t_m))))))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * (l_m * (l_m / (k_m * ((t_m * t_m) * (k_m * t_m)))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. unpow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lower-*.f64 51.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   3. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   10. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   12. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   13. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   17. lift-*.f64 55.3

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

6. Applied rewrites 55.3%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   5. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   6. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   7. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot \color{blue}{k}\right)} \]

   8. unpow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right)} \]

   9. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left({t}^{2} \cdot t\right) \cdot k\right)} \]

   10. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left({t}^{2} \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

   13. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   15. lower-*.f64 62.5

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right)}} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{k}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   18. lower-*.f64 62.5

       \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

8. Applied rewrites 62.5%

   \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot t\right)\right)}} \]
9. Add Preprocessing

Alternative 13: 61.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{l\_m}{k\_m \cdot \left(k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-154)
    (* (/ l_m (* k_m (* k_m (* (* t_m t_m) t_m)))) l_m)
    (* (/ l_m (* (* (* k_m k_m) (* t_m t_m)) t_m)) l_m))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 2.5e-154) {
		tmp = (l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m;
	} else {
		tmp = (l_m / (((k_m * k_m) * (t_m * t_m)) * t_m)) * l_m;
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-154) then
        tmp = (l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m
    else
        tmp = (l_m / (((k_m * k_m) * (t_m * t_m)) * t_m)) * l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 2.5e-154) {
		tmp = (l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m;
	} else {
		tmp = (l_m / (((k_m * k_m) * (t_m * t_m)) * t_m)) * l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 2.5e-154:
		tmp = (l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m
	else:
		tmp = (l_m / (((k_m * k_m) * (t_m * t_m)) * t_m)) * l_m
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-154)
		tmp = Float64(Float64(l_m / Float64(k_m * Float64(k_m * Float64(Float64(t_m * t_m) * t_m)))) * l_m);
	else
		tmp = Float64(Float64(l_m / Float64(Float64(Float64(k_m * k_m) * Float64(t_m * t_m)) * t_m)) * l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-154)
		tmp = (l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m;
	else
		tmp = (l_m / (((k_m * k_m) * (t_m * t_m)) * t_m)) * l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-154], N[(N[(l$95$m / N[(k$95$m * N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(N[(l$95$m / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000001e-154

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lower-*.f64 51.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      10. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      12. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      13. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 55.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. Applied rewrites 55.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 55.3

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 55.3%

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow3 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

      9. pow3 N/A

         \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

      11. lift-*.f64 59.6

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   10. Applied rewrites 59.6%

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   if 2.5000000000000001e-154 < k

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      11. lower-*.f64 51.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      9. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      10. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      12. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      13. pow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 55.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. Applied rewrites 55.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 55.3

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   8. Applied rewrites 55.3%

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      5. pow2 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      6. pow3 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

      7. unpow3 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

      8. pow2 N/A

         \[\leadsto \frac{\ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      12. pow2 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \cdot \ell \]

      14. pow2 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \ell \]

      15. lift-*.f64 58.0

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \ell \]

   10. Applied rewrites 58.0%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{l\_m}{k\_m \cdot \left(k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* (/ l_m (* k_m (* k_m (* (* t_m t_m) t_m)))) l_m)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m);
}

Fortran

l_m =     private
k_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m)
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m / Float64(k_m * Float64(k_m * Float64(Float64(t_m * t_m) * t_m)))) * l_m))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m / N[(k$95$m * N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. unpow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lower-*.f64 51.1

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   3. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   10. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   12. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   13. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   17. lift-*.f64 55.3

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

6. Applied rewrites 55.3%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 55.3

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

8. Applied rewrites 55.3%

   \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   5. pow3 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   9. pow3 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   11. lift-*.f64 59.6

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

10. Applied rewrites 59.6%

    \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(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))))