Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 82.2%

Time: 9.1s

Alternatives: 18

Speedup: 12.5×

• 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: 54.1% 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: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(\sin k \cdot t\_m\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k))))
   (*
    t_s
    (if (<= l_m 5.2e-162)
      (/ 2.0 (* (* t_2 k) (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (if (<= l_m 2.4e+149)
        (/
         2.0
         (*
          (/
           (fma (pow (sin k) 2.0) (* k k) (* (pow (* (sin k) t_m) 2.0) 2.0))
           (* (cos k) (* l_m l_m)))
          t_m))
        (/ 2.0 (* (* t_2 (tan k)) 2.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k);
	double tmp;
	if (l_m <= 5.2e-162) {
		tmp = 2.0 / ((t_2 * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (l_m <= 2.4e+149) {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (pow((sin(k) * t_m), 2.0) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((t_2 * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k))
	tmp = 0.0
	if (l_m <= 5.2e-162)
		tmp = Float64(2.0 / Float64(Float64(t_2 * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (l_m <= 2.4e+149)
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64((Float64(sin(k) * t_m) ^ 2.0) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 5.2e-162], N[(2.0 / N[(N[(t$95$2 * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2.4e+149], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 5.1999999999999999e-162

   1. Initial program 47.2%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 6.9

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

   7. Applied rewrites 6.9%

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

   if 5.1999999999999999e-162 < l < 2.40000000000000012e149

   1. Initial program 70.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   7. Applied rewrites 94.1%

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

   if 2.40000000000000012e149 < l

   1. Initial program 37.4%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 26.2

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

   7. Applied rewrites 26.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(\sin k \cdot t\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(t\_m \cdot t\_m\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{t\_2}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (* t_m t_m) t_m)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
         2e+139)
      (/
       (/
        2.0
        (*
         (/ (/ t_2 l_m) l_m)
         (*
          (fma
           (fma 0.08611111111111111 (* k k) 0.16666666666666666)
           (* k k)
           1.0)
          (* k k))))
       (fma (/ k t_m) (/ k t_m) 2.0))
      (* l_m (/ l_m (* (* k k) t_2)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m * t_m) * t_m;
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+139) {
		tmp = (2.0 / (((t_2 / l_m) / l_m) * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else {
		tmp = l_m * (l_m / ((k * k) * t_2));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m * t_m) * t_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+139)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(t_2 / l_m) / l_m) * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * k) * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+139], N[(N[(2.0 / N[(N[(N[(t$95$2 / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.00000000000000007e139

   1. Initial program 79.0%

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

   4. Applied rewrites 74.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.6

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

   6. Applied rewrites 74.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      13. lift-*.f64 70.8

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   9. Applied rewrites 70.8%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1} \]

       4. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1} \]

       5. associate-+l+ N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

       10. lift-/.f64 70.8

           \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   11. Applied rewrites 70.8%

       \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 2.00000000000000007e139 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 14.6%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 22.4

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

   5. Applied rewrites 22.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 29.0

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

   7. Applied rewrites 29.0%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 29.0

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

   9. Applied rewrites 29.0%

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

Alternative 3: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k))))
   (*
    t_s
    (if (<= (* l_m l_m) 0.0)
      (/ 2.0 (* (* t_2 k) (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (if (<= (* l_m l_m) 2e+296)
        (/
         2.0
         (*
          (/
           (fma 2.0 (pow (* (sin k) t_m) 2.0) (pow (* (sin k) k) 2.0))
           (* (cos k) (* l_m l_m)))
          t_m))
        (/ 2.0 (* (* t_2 (tan k)) 2.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k);
	double tmp;
	if ((l_m * l_m) <= 0.0) {
		tmp = 2.0 / ((t_2 * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if ((l_m * l_m) <= 2e+296) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t_m), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((t_2 * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k))
	tmp = 0.0
	if (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(t_2 * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (Float64(l_m * l_m) <= 2e+296)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t_m) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], N[(2.0 / N[(N[(t$95$2 * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+296], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 50.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 16.5

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

   7. Applied rewrites 16.5%

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

   if 0.0 < (*.f64 l l) < 1.99999999999999996e296

   1. Initial program 63.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.0%

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

   if 1.99999999999999996e296 < (*.f64 l l)

   1. Initial program 31.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 50.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 11.6

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

   7. Applied rewrites 11.6%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := e^{\log t\_m \cdot 3 - \log l\_m \cdot 2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.7 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{2}{t\_2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;l\_m \leq 1.22 \cdot 10^{-156}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}}\\ \mathbf{elif}\;l\_m \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0)))))
   (*
    t_s
    (if (<= l_m 4.7e-265)
      (/
       (/
        2.0
        (*
         t_2
         (*
          (fma
           (fma 0.08611111111111111 (* k k) 0.16666666666666666)
           (* k k)
           1.0)
          (* k k))))
       (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))
      (if (<= l_m 1.22e-156)
        (* l_m (/ l_m (exp (fma (log t_m) 3.0 (* (log k) 2.0)))))
        (if (<= l_m 2.4e+149)
          (/
           2.0
           (*
            (/
             (fma 2.0 (pow (* (sin k) t_m) 2.0) (pow (* (sin k) k) 2.0))
             (* (cos k) (* l_m l_m)))
            t_m))
          (/ 2.0 (* (* (* t_2 (sin k)) (tan k)) 2.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = exp(((log(t_m) * 3.0) - (log(l_m) * 2.0)));
	double tmp;
	if (l_m <= 4.7e-265) {
		tmp = (2.0 / (t_2 * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)))) / ((pow((k / t_m), 2.0) + 1.0) + 1.0);
	} else if (l_m <= 1.22e-156) {
		tmp = l_m * (l_m / exp(fma(log(t_m), 3.0, (log(k) * 2.0))));
	} else if (l_m <= 2.4e+149) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t_m), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((t_2 * sin(k)) * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0)))
	tmp = 0.0
	if (l_m <= 4.7e-265)
		tmp = Float64(Float64(2.0 / Float64(t_2 * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k)))) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0));
	elseif (l_m <= 1.22e-156)
		tmp = Float64(l_m * Float64(l_m / exp(fma(log(t_m), 3.0, Float64(log(k) * 2.0)))));
	elseif (l_m <= 2.4e+149)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t_m) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * sin(k)) * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 4.7e-265], N[(N[(2.0 / N[(t$95$2 * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.22e-156], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2.4e+149], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if l < 4.69999999999999986e-265

   1. Initial program 44.1%

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

   4. Applied rewrites 48.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 48.4

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

   6. Applied rewrites 48.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      13. lift-*.f64 44.8

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   9. Applied rewrites 44.8%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\frac{\left(t \cdot t\right) \cdot t}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       6. pow3 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       7. pow-to-exp N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       8. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       9. pow-to-exp N/A

          \[\leadsto \frac{\frac{2}{\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       10. div-exp N/A

           \[\leadsto \frac{\frac{2}{\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       11. lower-exp.f64 N/A

           \[\leadsto \frac{\frac{2}{\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       12. lower--.f64 N/A

           \[\leadsto \frac{\frac{2}{e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{\frac{2}{e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       14. lower-log.f64 N/A

           \[\leadsto \frac{\frac{2}{e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{\frac{2}{e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       16. lower-log.f64 3.6

           \[\leadsto \frac{\frac{2}{e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   11. Applied rewrites 3.6%

       \[\leadsto \frac{\frac{2}{\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   if 4.69999999999999986e-265 < l < 1.21999999999999995e-156

   1. Initial program 64.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.5

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

   5. Applied rewrites 45.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 55.1

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

   7. Applied rewrites 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3} \cdot e^{\log k \cdot 2}} \]

      8. prod-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. lower-exp.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 13.1

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

   9. Applied rewrites 13.1%

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

   if 1.21999999999999995e-156 < l < 2.40000000000000012e149

   1. Initial program 71.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

   if 2.40000000000000012e149 < l

   1. Initial program 37.4%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 26.2

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

   7. Applied rewrites 26.2%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{2}{e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{-156}:\\ \;\;\;\;\ell \cdot \frac{\ell}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, \left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.5e-120)
    (/
     2.0
     (*
      (/
       (fma (pow (sin k) 2.0) (* k k) (* (* (* k t_m) (* k t_m)) 2.0))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 4.6e+23)
      (/
       2.0
       (*
        (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k))
        (* (tan k) (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))))
      (* l_m (/ l_m (exp (fma (log t_m) 3.0 (* (log k) 2.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.5e-120) {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (((k * t_m) * (k * t_m)) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 4.6e+23) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l_m) / l_m) * sin(k)) * (tan(k) * ((pow((k / t_m), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = l_m * (l_m / exp(fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9.5e-120)
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 4.6e+23)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * Float64(tan(k) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0))));
	else
		tmp = Float64(l_m * Float64(l_m / exp(fma(log(t_m), 3.0, Float64(log(k) * 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 9.5e-120], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e+23], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.49999999999999937e-120

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(k \cdot t\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.9%

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

       1. lift-pow.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 74.9

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

   12. Applied rewrites 74.9%

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

   if 9.49999999999999937e-120 < t < 4.6000000000000001e23

   1. Initial program 68.7%

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

   4. Applied rewrites 79.1%

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

   if 4.6000000000000001e23 < t

   1. Initial program 60.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 58.2

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

   7. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3} \cdot e^{\log k \cdot 2}} \]

      8. prod-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. lower-exp.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 34.4

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

   9. Applied rewrites 34.4%

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

Alternative 6: 37.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-96}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, \left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.1e-96)
    (* l_m (/ l_m (exp (fma (log t_m) 3.0 (* (log k) 2.0)))))
    (/
     2.0
     (*
      (/
       (fma (pow (sin k) 2.0) (* k k) (* (* (* k t_m) (* k t_m)) 2.0))
       (* (cos k) (* l_m l_m)))
      t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.1e-96) {
		tmp = l_m * (l_m / exp(fma(log(t_m), 3.0, (log(k) * 2.0))));
	} else {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (((k * t_m) * (k * t_m)) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.1e-96)
		tmp = Float64(l_m * Float64(l_m / exp(fma(log(t_m), 3.0, Float64(log(k) * 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 1.1e-96], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.0999999999999999e-96

   1. Initial program 57.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.8

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

   5. Applied rewrites 48.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3} \cdot e^{\log k \cdot 2}} \]

      8. prod-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. lower-exp.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 9.9

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

   9. Applied rewrites 9.9%

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

   if 1.0999999999999999e-96 < k

   1. Initial program 41.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(k \cdot t\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.7%

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

       1. lift-pow.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 73.7

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

   12. Applied rewrites 73.7%

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

Alternative 7: 55.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}}{l\_m} \cdot \left(\sin k \cdot \tan k\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-120)
    (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (* l_m l_m))) t_m))
    (if (<= t_m 4.6e+23)
      (/
       (/ 2.0 (* (/ (* (* t_m t_m) (/ t_m l_m)) l_m) (* (sin k) (tan k))))
       (fma (/ k t_m) (/ k t_m) 2.0))
      (* l_m (/ l_m (exp (fma (log t_m) 3.0 (* (log k) 2.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.3e-120) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 4.6e+23) {
		tmp = (2.0 / ((((t_m * t_m) * (t_m / l_m)) / l_m) * (sin(k) * tan(k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else {
		tmp = l_m * (l_m / exp(fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.3e-120)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 4.6e+23)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l_m)) / l_m) * Float64(sin(k) * tan(k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	else
		tmp = Float64(l_m * Float64(l_m / exp(fma(log(t_m), 3.0, Float64(log(k) * 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-120], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e+23], N[(N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.29999999999999986e-120

   1. Initial program 47.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 61.6

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

   8. Applied rewrites 61.6%

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

   if 2.29999999999999986e-120 < t < 4.6000000000000001e23

   1. Initial program 66.0%

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

   4. Applied rewrites 72.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 72.8

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

   6. Applied rewrites 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 77.2

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

   8. Applied rewrites 77.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 77.2

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

   10. Applied rewrites 77.2%

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

   if 4.6000000000000001e23 < t

   1. Initial program 60.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 58.2

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

   7. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3} \cdot e^{\log k \cdot 2}} \]

      8. prod-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. lower-exp.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 34.4

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

   9. Applied rewrites 34.4%

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

Alternative 8: 52.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}}{l\_m} \cdot \left(\sin k \cdot \tan k\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{e^{\mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-157)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 4.6e+23)
      (/
       (/ 2.0 (* (/ (* (* t_m t_m) (/ t_m l_m)) l_m) (* (sin k) (tan k))))
       (fma (/ k t_m) (/ k t_m) 2.0))
      (* l_m (/ l_m (exp (fma (log t_m) 3.0 (* (log k) 2.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.8e-157) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 4.6e+23) {
		tmp = (2.0 / ((((t_m * t_m) * (t_m / l_m)) / l_m) * (sin(k) * tan(k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else {
		tmp = l_m * (l_m / exp(fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.8e-157)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 4.6e+23)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l_m)) / l_m) * Float64(sin(k) * tan(k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	else
		tmp = Float64(l_m * Float64(l_m / exp(fma(log(t_m), 3.0, Float64(log(k) * 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-157], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.6e+23], N[(N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.8e-157

   1. Initial program 48.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.8e-157 < t < 4.6000000000000001e23

   1. Initial program 56.4%

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

   4. Applied rewrites 61.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.7

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

   6. Applied rewrites 61.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 70.8

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

   8. Applied rewrites 70.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 70.8

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

   10. Applied rewrites 70.8%

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

   if 4.6000000000000001e23 < t

   1. Initial program 60.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 58.2

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

   7. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3} \cdot e^{\log k \cdot 2}} \]

      8. prod-exp N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. lower-exp.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 34.4

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

   9. Applied rewrites 34.4%

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

Alternative 9: 69.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{2}{\frac{\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}}{l\_m} \cdot \left(\sin k \cdot \tan k\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-157)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 2.4e+88)
      (/
       (/ 2.0 (* (/ (* (* t_m t_m) (/ t_m l_m)) l_m) (* (sin k) (tan k))))
       (fma (/ k t_m) (/ k t_m) 2.0))
      (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.8e-157) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 2.4e+88) {
		tmp = (2.0 / ((((t_m * t_m) * (t_m / l_m)) / l_m) * (sin(k) * tan(k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.8e-157)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 2.4e+88)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l_m)) / l_m) * Float64(sin(k) * tan(k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-157], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+88], N[(N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.8e-157

   1. Initial program 48.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.8e-157 < t < 2.3999999999999999e88

   1. Initial program 54.0%

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

   4. Applied rewrites 63.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 63.4

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

   6. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 69.7

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

   8. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 69.7

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

   10. Applied rewrites 69.7%

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

   if 2.3999999999999999e88 < t

   1. Initial program 64.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 82.7

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

   8. Applied rewrites 82.7%

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

Alternative 10: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \left(\sin k \cdot \tan k\right)}}{\frac{k \cdot k}{t\_m \cdot t\_m} + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.1e-120)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 1.9e+91)
      (/
       (/ 2.0 (* (/ (/ (* (* t_m t_m) t_m) l_m) l_m) (* (sin k) (tan k))))
       (+ (/ (* k k) (* t_m t_m)) 2.0))
      (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.1e-120) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 1.9e+91) {
		tmp = (2.0 / (((((t_m * t_m) * t_m) / l_m) / l_m) * (sin(k) * tan(k)))) / (((k * k) / (t_m * t_m)) + 2.0);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 6.1e-120)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 1.9e+91)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * Float64(sin(k) * tan(k)))) / Float64(Float64(Float64(k * k) / Float64(t_m * t_m)) + 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.1e-120], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+91], N[(N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.1e-120

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.8

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 6.1e-120 < t < 1.8999999999999999e91

   1. Initial program 59.7%

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

   4. Applied rewrites 70.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.8

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 66.0

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

   9. Applied rewrites 66.0%

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

   if 1.8999999999999999e91 < t

   1. Initial program 66.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

Alternative 11: 67.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 0.0017:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \left(t\_m \cdot \frac{t\_m}{l\_m}\right)}{l\_m} \cdot \left(k \cdot \tan k\right)}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-157)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 0.0017)
      (/
       (/ 2.0 (* (/ (* t_m (* t_m (/ t_m l_m))) l_m) (* k (tan k))))
       (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))
      (if (<= t_m 1.85e+99)
        (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
        (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.8e-157) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 0.0017) {
		tmp = (2.0 / (((t_m * (t_m * (t_m / l_m))) / l_m) * (k * tan(k)))) / ((pow((k / t_m), 2.0) + 1.0) + 1.0);
	} else if (t_m <= 1.85e+99) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.8e-157)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 0.0017)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(t_m * Float64(t_m * Float64(t_m / l_m))) / l_m) * Float64(k * tan(k)))) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0));
	elseif (t_m <= 1.85e+99)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-157], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0017], N[(N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e+99], N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.8e-157

   1. Initial program 48.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.8e-157 < t < 0.00169999999999999991

   1. Initial program 50.7%

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

   4. Applied rewrites 57.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 57.2

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

   6. Applied rewrites 57.2%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 50.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r/ N/A

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

       6. pow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-/.f64 61.3

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

   11. Applied rewrites 61.3%

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

   if 0.00169999999999999991 < t < 1.85000000000000005e99

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.4

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

   5. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 49.8

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

   7. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

   if 1.85000000000000005e99 < t

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

Alternative 12: 67.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 0.00106:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \left(t\_m \cdot \frac{t\_m}{l\_m}\right)}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-157)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 0.00106)
      (/
       (/
        2.0
        (*
         (/ (* t_m (* t_m (/ t_m l_m))) l_m)
         (*
          (fma
           (fma 0.08611111111111111 (* k k) 0.16666666666666666)
           (* k k)
           1.0)
          (* k k))))
       (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))
      (if (<= t_m 1.85e+99)
        (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
        (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.8e-157) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 0.00106) {
		tmp = (2.0 / (((t_m * (t_m * (t_m / l_m))) / l_m) * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)))) / ((pow((k / t_m), 2.0) + 1.0) + 1.0);
	} else if (t_m <= 1.85e+99) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.8e-157)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 0.00106)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(t_m * Float64(t_m * Float64(t_m / l_m))) / l_m) * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k)))) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0));
	elseif (t_m <= 1.85e+99)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-157], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.00106], N[(N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e+99], N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.8e-157

   1. Initial program 48.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.8e-157 < t < 0.00105999999999999996

   1. Initial program 50.7%

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

   4. Applied rewrites 57.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 57.2

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

   6. Applied rewrites 57.2%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      13. lift-*.f64 50.1

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   9. Applied rewrites 50.1%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\frac{\left(t \cdot t\right) \cdot t}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{2}} \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{2} \cdot \frac{t}{\ell}}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       6. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot \frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       7. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot \frac{t}{\ell}\right)}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot \frac{t}{\ell}\right)}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot \frac{t}{\ell}\right)}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

       10. lift-/.f64 61.3

           \[\leadsto \frac{\frac{2}{\frac{t \cdot \left(t \cdot \color{blue}{\frac{t}{\ell}}\right)}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   11. Applied rewrites 61.3%

       \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot \frac{t}{\ell}\right)}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   if 0.00105999999999999996 < t < 1.85000000000000005e99

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.4

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

   5. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 49.8

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

   7. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

   if 1.85000000000000005e99 < t

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

Alternative 13: 67.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 0.00106:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \left(k \cdot k\right)}}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.1e-120)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 0.00106)
      (/
       (/ 2.0 (* (/ (/ (* (* t_m t_m) t_m) l_m) l_m) (* k k)))
       (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))
      (if (<= t_m 1.85e+99)
        (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
        (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.1e-120) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 0.00106) {
		tmp = (2.0 / (((((t_m * t_m) * t_m) / l_m) / l_m) * (k * k))) / ((pow((k / t_m), 2.0) + 1.0) + 1.0);
	} else if (t_m <= 1.85e+99) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 6.1e-120)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 0.00106)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * Float64(k * k))) / Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0));
	elseif (t_m <= 1.85e+99)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.1e-120], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.00106], N[(N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e+99], N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 6.1e-120

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.8

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 6.1e-120 < t < 0.00105999999999999996

   1. Initial program 64.1%

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

   4. Applied rewrites 73.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.4

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 63.3

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

   9. Applied rewrites 63.3%

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

   if 0.00105999999999999996 < t < 1.85000000000000005e99

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.4

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

   5. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 49.8

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

   7. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

   if 1.85000000000000005e99 < t

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

Alternative 14: 67.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t\_m \cdot t\_m, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 0.00106:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.1e-120)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma -0.6666666666666666 (* t_m t_m) 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) (* l_m l_m)))
      t_m))
    (if (<= t_m 0.00106)
      (/
       (/
        2.0
        (*
         (/ (/ (* (* t_m t_m) t_m) l_m) l_m)
         (*
          (fma
           (fma 0.08611111111111111 (* k k) 0.16666666666666666)
           (* k k)
           1.0)
          (* k k))))
       (fma (/ k t_m) (/ k t_m) 2.0))
      (if (<= t_m 1.85e+99)
        (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
        (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.1e-120) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t_m * t_m), 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t_m);
	} else if (t_m <= 0.00106) {
		tmp = (2.0 / (((((t_m * t_m) * t_m) / l_m) / l_m) * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else if (t_m <= 1.85e+99) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 6.1e-120)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t_m * t_m), 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	elseif (t_m <= 0.00106)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	elseif (t_m <= 1.85e+99)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.1e-120], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.00106], N[(N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e+99], N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 6.1e-120

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      17. lift-*.f64 49.8

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 6.1e-120 < t < 0.00105999999999999996

   1. Initial program 64.1%

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

   4. Applied rewrites 73.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.4

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      13. lift-*.f64 63.0

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   9. Applied rewrites 63.0%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1} \]

       4. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1} \]

       5. associate-+l+ N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

       10. lift-/.f64 63.0

           \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   11. Applied rewrites 63.0%

       \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 0.00105999999999999996 < t < 1.85000000000000005e99

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.4

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

   5. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 49.8

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

   7. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.8

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

   9. Applied rewrites 61.8%

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

   if 1.85000000000000005e99 < t

   1. Initial program 66.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

Alternative 15: 65.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.6 \cdot 10^{-113}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 0.00106:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m))))
   (*
    t_s
    (if (<= t_m 8.6e-113)
      t_2
      (if (<= t_m 0.00106)
        (/
         (/
          2.0
          (*
           (/ (/ (* (* t_m t_m) t_m) l_m) l_m)
           (*
            (fma
             (fma 0.08611111111111111 (* k k) 0.16666666666666666)
             (* k k)
             1.0)
            (* k k))))
         (fma (/ k t_m) (/ k t_m) 2.0))
        (if (<= t_m 1.85e+99)
          (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	double tmp;
	if (t_m <= 8.6e-113) {
		tmp = t_2;
	} else if (t_m <= 0.00106) {
		tmp = (2.0 / (((((t_m * t_m) * t_m) / l_m) / l_m) * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else if (t_m <= 1.85e+99) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m))
	tmp = 0.0
	if (t_m <= 8.6e-113)
		tmp = t_2;
	elseif (t_m <= 0.00106)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	elseif (t_m <= 1.85e+99)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.6e-113], t$95$2, If[LessEqual[t$95$m, 0.00106], N[(N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e+99], N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.6000000000000001e-113 or 1.85000000000000005e99 < t

   1. Initial program 51.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      8. lift-*.f64 65.4

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   8. Applied rewrites 65.4%

      \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   if 8.6000000000000001e-113 < t < 0.00105999999999999996

   1. Initial program 64.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. unpow3 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{2}} \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{2} \cdot t}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. lower-*.f64 73.4

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   6. Applied rewrites 73.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      13. lift-*.f64 63.0

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   9. Applied rewrites 63.0%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1} \]

       4. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1} \]

       5. associate-+l+ N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

       10. lift-/.f64 63.0

           \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   11. Applied rewrites 63.0%

       \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 0.00105999999999999996 < t < 1.85000000000000005e99

   1. Initial program 55.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      6. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      8. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 49.8

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 49.8%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{{t}^{3}}\right)} \]

      7. lift-pow.f64 61.8

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{\color{blue}{3}}\right)} \]

   9. Applied rewrites 61.8%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 63.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.6 \cdot 10^{-113}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 0.00106:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m))))
   (*
    t_s
    (if (<= t_m 8.6e-113)
      t_2
      (if (<= t_m 0.00106)
        (/
         (/
          2.0
          (*
           (/ (/ (* (* t_m t_m) t_m) l_m) l_m)
           (*
            (fma
             (fma 0.08611111111111111 (* k k) 0.16666666666666666)
             (* k k)
             1.0)
            (* k k))))
         (fma (/ k t_m) (/ k t_m) 2.0))
        (if (<= t_m 1.7e+99)
          (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
          t_2))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	double tmp;
	if (t_m <= 8.6e-113) {
		tmp = t_2;
	} else if (t_m <= 0.00106) {
		tmp = (2.0 / (((((t_m * t_m) * t_m) / l_m) / l_m) * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)))) / fma((k / t_m), (k / t_m), 2.0);
	} else if (t_m <= 1.7e+99) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m))
	tmp = 0.0
	if (t_m <= 8.6e-113)
		tmp = t_2;
	elseif (t_m <= 0.00106)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k)))) / fma(Float64(k / t_m), Float64(k / t_m), 2.0));
	elseif (t_m <= 1.7e+99)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.6e-113], t$95$2, If[LessEqual[t$95$m, 0.00106], N[(N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+99], N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.6000000000000001e-113 or 1.69999999999999992e99 < t

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 49.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 49.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 49.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 63.5

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 63.5%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]

   if 8.6000000000000001e-113 < t < 0.00105999999999999996

   1. Initial program 64.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. unpow3 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{2}} \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{{t}^{2} \cdot t}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. lower-*.f64 73.4

         \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   6. Applied rewrites 73.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot \color{blue}{{k}^{2}}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {\color{blue}{k}}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      6. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

      13. lift-*.f64 63.0

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]

   9. Applied rewrites 63.0%

      \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left({\color{blue}{\left(\frac{k}{t}\right)}}^{2} + 1\right) + 1} \]

       4. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 1\right) + 1} \]

       5. associate-+l+ N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\color{blue}{\frac{k}{t}}, \frac{k}{t}, 2\right)} \]

       10. lift-/.f64 63.0

           \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\mathsf{fma}\left(\frac{k}{t}, \color{blue}{\frac{k}{t}}, 2\right)} \]

   11. Applied rewrites 63.0%

       \[\leadsto \frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   if 0.00105999999999999996 < t < 1.69999999999999992e99

   1. Initial program 55.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 48.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      6. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      8. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 49.8

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 49.8%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{{t}^{3}}\right)} \]

      7. lift-pow.f64 61.8

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{\color{blue}{3}}\right)} \]

   9. Applied rewrites 61.8%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / (k * (k * pow(t_m, 3.0)))));
}

Fortran

l_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)
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
    code = t_s * (l_m * (l_m / (k * (k * (t_m ** 3.0d0)))))
end function

Java

l_m = Math.abs(l);
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) {
	return t_s * (l_m * (l_m / (k * (k * Math.pow(t_m, 3.0)))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / (k * (k * math.pow(t_m, 3.0)))))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0))))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / (k * (k * (t_m ^ 3.0)))));
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. lift-pow.f64 47.9

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 47.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   6. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   8. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   9. lower-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   13. lift-*.f64 51.3

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

7. Applied rewrites 51.3%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   3. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{{t}^{3}}\right)} \]

   7. lift-pow.f64 57.8

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{\color{blue}{3}}\right)} \]

9. Applied rewrites 57.8%

   \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]
10. Add Preprocessing

Alternative 18: 54.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* l_m (/ l_m (* (* k k) (* (* t_m t_m) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
}

Fortran

l_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)
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
    code = t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))))
end function

Java

l_m = Math.abs(l);
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) {
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. lift-pow.f64 47.9

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 47.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   6. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   8. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   9. lower-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   13. lift-*.f64 51.3

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

7. Applied rewrites 51.3%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. lift-*.f64 51.3

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

9. Applied rewrites 51.3%

   \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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))))