Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 85.0%

Time: 7.8s

Alternatives: 24

Speedup: 5.4×

• 27.1% 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: 54.7% 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: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m} \cdot \mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= l_m 3.5e+214)
    (/
     2.0
     (/
      (*
       (/ t_m l_m)
       (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)))
      (* (cos k) l_m)))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (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 tmp;
	if (l_m <= 3.5e+214) {
		tmp = 2.0 / (((t_m / l_m) * fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0))) / (cos(k) * l_m));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * 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)
	tmp = 0.0
	if (l_m <= 3.5e+214)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) * fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0))) / Float64(cos(k) * l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * 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_] := N[(t$95$s * If[LessEqual[l$95$m, 3.5e+214], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e214

   1. Initial program 58.4%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.7%

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

   6. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

   10. Applied rewrites 86.8%

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

   if 3.5e214 < l

   1. Initial program 33.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      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 74.8

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

   6. Applied rewrites 74.8%

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

Alternative 2: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left({\left(k \cdot t\_m\right)}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, k \cdot k, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        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)))
       5e+219)
    (/
     2.0
     (*
      (/ t_m (* l_m l_m))
      (* (pow (* k t_m) 2.0) (fma 0.3333333333333333 (* k k) 2.0))))
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (*
       (fma
        (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
        (* k k)
        (* (* t_m t_m) 2.0))
       (* k k)))))))

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 ((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))) <= 5e+219) {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (pow((k * t_m), 2.0) * fma(0.3333333333333333, (k * k), 2.0)));
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	}
	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 (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))) <= 5e+219)
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64((Float64(k * t_m) ^ 2.0) * fma(0.3333333333333333, Float64(k * k), 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	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[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], 5e+219], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.3333333333333333 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $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)))) < 5e219

   1. Initial program 77.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 86.8%

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

   6. Applied rewrites 86.3%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 35.7%

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

   10. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. unpow-prod-down N/A

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

       4. lift-pow.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 78.2

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

   12. Applied rewrites 78.2%

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

   if 5e219 < (/.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 35.1%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 52.5%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 57.3

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

       6. unpow-prod-down 57.3

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

       7. *-commutative 57.3

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

       8. *-commutative 57.3

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

       9. *-commutative 57.3

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

       10. unpow-prod-down 57.3

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

   11. Applied rewrites 60.3%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{l\_m \cdot l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        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)))
       5e+219)
    (/ 2.0 (* (/ (* (pow (* k t_m) 2.0) 2.0) (* l_m l_m)) t_m))
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (*
       (fma
        (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
        (* k k)
        (* (* t_m t_m) 2.0))
       (* k k)))))))

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 ((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))) <= 5e+219) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) / (l_m * l_m)) * t_m);
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	}
	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 (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))) <= 5e+219)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / Float64(l_m * l_m)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	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[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], 5e+219], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $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)))) < 5e219

   1. Initial program 77.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 86.8%

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

   6. Applied rewrites 86.8%

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

   7. Taylor expanded in k around 0

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

      1. unpow-prod-down N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in k around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-pow.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 78.7

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

   12. Applied rewrites 78.7%

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

   if 5e219 < (/.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 35.1%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 52.5%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 57.3

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

       6. unpow-prod-down 57.3

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

       7. *-commutative 57.3

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

       8. *-commutative 57.3

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

       9. *-commutative 57.3

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

       10. unpow-prod-down 57.3

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

   11. Applied rewrites 60.3%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+219}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        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)))
       5e+219)
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m))
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (*
       (fma
        (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
        (* k k)
        (* (* t_m t_m) 2.0))
       (* k k)))))))

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 ((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))) <= 5e+219) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	}
	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 (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))) <= 5e+219)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	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[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], 5e+219], 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[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $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)))) < 5e219

   1. Initial program 77.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) \cdot t} \]
   6. 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 78.7

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

   7. Applied rewrites 78.7%

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

   if 5e219 < (/.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 35.1%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 52.5%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 57.3

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

       6. unpow-prod-down 57.3

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

       7. *-commutative 57.3

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

       8. *-commutative 57.3

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

       9. *-commutative 57.3

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

       10. unpow-prod-down 57.3

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

   11. Applied rewrites 60.3%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot l\_m}}{l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= l_m 3.5e+214)
    (/
     2.0
     (*
      (/
       (/
        (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))
        (* (cos k) l_m))
       l_m)
      t_m))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (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 tmp;
	if (l_m <= 3.5e+214) {
		tmp = 2.0 / (((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / (cos(k) * l_m)) / l_m) * t_m);
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * 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)
	tmp = 0.0
	if (l_m <= 3.5e+214)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * l_m)) / l_m) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * 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_] := N[(t$95$s * If[LessEqual[l$95$m, 3.5e+214], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.5e214

   1. Initial program 58.4%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.7%

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

   6. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 85.2%

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

   if 3.5e214 < l

   1. Initial program 33.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      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 74.8

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

   6. Applied rewrites 74.8%

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

Alternative 6: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= l_m 5.5e+120)
    (/
     2.0
     (*
      (/
       (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))
       (* (* (cos k) l_m) l_m))
      t_m))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (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 tmp;
	if (l_m <= 5.5e+120) {
		tmp = 2.0 / ((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / ((cos(k) * l_m) * l_m)) * t_m);
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * 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)
	tmp = 0.0
	if (l_m <= 5.5e+120)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / Float64(Float64(cos(k) * l_m) * l_m)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * 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_] := N[(t$95$s * If[LessEqual[l$95$m, 5.5e+120], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.50000000000000003e120

   1. Initial program 62.6%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 82.2%

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

   6. Applied rewrites 82.2%

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

   if 5.50000000000000003e120 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      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 71.2

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

   6. Applied rewrites 71.2%

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

Alternative 7: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= l_m 5.5e+120)
    (/
     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
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (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 tmp;
	if (l_m <= 5.5e+120) {
		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 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * 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)
	tmp = 0.0
	if (l_m <= 5.5e+120)
		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(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * 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_] := N[(t$95$s * If[LessEqual[l$95$m, 5.5e+120], 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[(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[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.50000000000000003e120

   1. Initial program 62.6%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 82.2%

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

   if 5.50000000000000003e120 < l

   1. Initial program 36.2%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      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 71.2

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

   6. Applied rewrites 71.2%

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

Alternative 8: 64.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq \infty:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left(\left(t\_2 \cdot k\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(t\_2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

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
         (fma
          (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
          (* k k)
          (* (* t_m t_m) 2.0))))
   (*
    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)))
         INFINITY)
      (/ 2.0 (* (/ t_m (* l_m l_m)) (* (* t_2 k) k)))
      (/ 2.0 (* (/ (/ t_m l_m) l_m) (* t_2 (* k k))))))))

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 = fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0));
	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))) <= ((double) INFINITY)) {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((t_2 * k) * k));
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (t_2 * (k * k)));
	}
	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 = fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0))
	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))) <= Inf)
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(Float64(t_2 * k) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(t_2 * Float64(k * k))));
	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[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $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], Infinity], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$2 * N[(k * k), $MachinePrecision]), $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)))) < +inf.0

   1. Initial program 82.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 89.0%

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 53.2%

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

   11. Applied rewrites 76.5%

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

   if +inf.0 < (/.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 0.0%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 43.7%

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

   6. Applied rewrites 43.7%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 28.2%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 35.5

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

       6. unpow-prod-down 35.5

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

       7. *-commutative 35.5

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

       8. *-commutative 35.5

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

       9. *-commutative 35.5

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

       10. unpow-prod-down 35.5

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

   11. Applied rewrites 40.3%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{l\_m \cdot l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(\mathsf{fma}\left(0.3333333333333333, k \cdot k, 2\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (* l_m l_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)))
         4e+302)
      (/
       2.0
       (*
        t_2
        (* (* (fma 0.3333333333333333 (* k k) 2.0) (* t_m t_m)) (* k k))))
      (/ 2.0 (* t_2 (* (* k k) (* k k))))))))

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 / (l_m * l_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))) <= 4e+302) {
		tmp = 2.0 / (t_2 * ((fma(0.3333333333333333, (k * k), 2.0) * (t_m * t_m)) * (k * k)));
	} else {
		tmp = 2.0 / (t_2 * ((k * k) * (k * k)));
	}
	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(t_m / Float64(l_m * l_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))) <= 4e+302)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(fma(0.3333333333333333, Float64(k * k), 2.0) * Float64(t_m * t_m)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k * k) * Float64(k * k))));
	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[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $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], 4e+302], N[(2.0 / N[(t$95$2 * N[(N[(N[(0.3333333333333333 * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $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)))) < 4.0000000000000003e302

   1. Initial program 77.8%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 86.3%

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 35.9%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 65.2

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

   12. Applied rewrites 65.2%

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

   if 4.0000000000000003e302 < (/.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 34.4%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 52.5%

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

   10. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 53.7

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

   12. Applied rewrites 53.7%

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

Alternative 10: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{l\_m \cdot l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (* l_m l_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)))
         4e+302)
      (/ 2.0 (* t_2 (* (* (* t_m t_m) 2.0) (* k k))))
      (/ 2.0 (* t_2 (* (* k k) (* k k))))))))

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 / (l_m * l_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))) <= 4e+302) {
		tmp = 2.0 / (t_2 * (((t_m * t_m) * 2.0) * (k * k)));
	} else {
		tmp = 2.0 / (t_2 * ((k * k) * (k * k)));
	}
	return t_s * tmp;
}

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
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m / (l_m * l_m)
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+302) then
        tmp = 2.0d0 / (t_2 * (((t_m * t_m) * 2.0d0) * (k * k)))
    else
        tmp = 2.0d0 / (t_2 * ((k * k) * (k * k)))
    end if
    code = t_s * tmp
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) {
	double t_2 = t_m / (l_m * l_m);
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+302) {
		tmp = 2.0 / (t_2 * (((t_m * t_m) * 2.0) * (k * k)));
	} else {
		tmp = 2.0 / (t_2 * ((k * k) * (k * k)));
	}
	return t_s * tmp;
}

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):
	t_2 = t_m / (l_m * l_m)
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+302:
		tmp = 2.0 / (t_2 * (((t_m * t_m) * 2.0) * (k * k)))
	else:
		tmp = 2.0 / (t_2 * ((k * k) * (k * k)))
	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(t_m / Float64(l_m * l_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))) <= 4e+302)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(Float64(t_m * t_m) * 2.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k * k) * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = t_m / (l_m * l_m);
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+302)
		tmp = 2.0 / (t_2 * (((t_m * t_m) * 2.0) * (k * k)));
	else
		tmp = 2.0 / (t_2 * ((k * k) * (k * k)));
	end
	tmp_2 = 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[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $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], 4e+302], N[(2.0 / N[(t$95$2 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $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)))) < 4.0000000000000003e302

   1. Initial program 77.8%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 86.3%

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 35.9%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 65.1

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

   12. Applied rewrites 65.1%

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

   if 4.0000000000000003e302 < (/.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 34.4%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 52.5%

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

   10. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 53.7

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

   12. Applied rewrites 53.7%

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

Alternative 11: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+302}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        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)))
       4e+302)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (/ t_m (* l_m l_m)) (* (* k k) (* k k)))))))

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 ((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))) <= 4e+302) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((k * k) * (k * k)));
	}
	return t_s * tmp;
}

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
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+302) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / ((t_m / (l_m * l_m)) * ((k * k) * (k * k)))
    end if
    code = t_s * tmp
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) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+302) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((k * k) * (k * k)));
	}
	return t_s * tmp;
}

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):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+302:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((k * k) * (k * k)))
	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 (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))) <= 4e+302)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(Float64(k * k) * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+302)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((k * k) * (k * k)));
	end
	tmp_2 = 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[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], 4e+302], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $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)))) < 4.0000000000000003e302

   1. Initial program 77.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 63.4

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

   4. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. 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 63.4

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

   6. Applied rewrites 63.4%

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

   if 4.0000000000000003e302 < (/.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 34.4%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 52.5%

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

   10. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 53.7

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

   12. Applied rewrites 53.7%

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

Alternative 12: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

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-9)
    (/ 2.0 (* (/ (/ t_m l_m) l_m) (/ (pow (* (sin k) k) 2.0) (cos k))))
    (/ 2.0 (* (/ t_m (* l_m l_m)) (* (pow (* k t_m) 2.0) 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-9) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (pow((sin(k) * k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (pow((k * t_m), 2.0) * 2.0));
	}
	return t_s * tmp;
}

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
    real(8) :: tmp
    if (t_m <= 1.8d-9) then
        tmp = 2.0d0 / (((t_m / l_m) / l_m) * (((sin(k) * k) ** 2.0d0) / cos(k)))
    else
        tmp = 2.0d0 / ((t_m / (l_m * l_m)) * (((k * t_m) ** 2.0d0) * 2.0d0))
    end if
    code = t_s * tmp
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) {
	double tmp;
	if (t_m <= 1.8e-9) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (Math.pow((Math.sin(k) * k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (Math.pow((k * t_m), 2.0) * 2.0));
	}
	return t_s * tmp;
}

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):
	tmp = 0
	if t_m <= 1.8e-9:
		tmp = 2.0 / (((t_m / l_m) / l_m) * (math.pow((math.sin(k) * k), 2.0) / math.cos(k)))
	else:
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (math.pow((k * t_m), 2.0) * 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-9)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64((Float64(k * t_m) ^ 2.0) * 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.8e-9)
		tmp = 2.0 / (((t_m / l_m) / l_m) * (((sin(k) * k) ^ 2.0) / cos(k)));
	else
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (((k * t_m) ^ 2.0) * 2.0));
	end
	tmp_2 = 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-9], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8e-9

   1. Initial program 42.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 71.8%

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.5

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in t around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. lower-/.f64 N/A

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

       7. *-commutative N/A

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

       8. unpow-prod-down N/A

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

       9. lift-sin.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-pow.f64 N/A

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

       12. lift-cos.f64 76.6

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

   11. Applied rewrites 76.6%

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

   if 1.8e-9 < 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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.3%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 71.5

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

   9. Applied rewrites 71.5%

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

Alternative 13: 68.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2} \cdot t\_m}{\left(\cos k \cdot l\_m\right) \cdot l\_m}}\\ \end{array} \end{array} \]

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 1.95e-119)
    (/ 2.0 (* (/ t_m (* l_m l_m)) (* (pow (* k t_m) 2.0) 2.0)))
    (if (<= k 3e+15)
      (/
       2.0
       (*
        (/ (/ t_m l_m) l_m)
        (*
         (fma
          (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
          (* k k)
          (* (* t_m t_m) 2.0))
         (* k k))))
      (/ 2.0 (/ (* (pow (* (sin k) k) 2.0) t_m) (* (* (cos k) l_m) l_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 <= 1.95e-119) {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (pow((k * t_m), 2.0) * 2.0));
	} else if (k <= 3e+15) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) * t_m) / ((cos(k) * l_m) * l_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 <= 1.95e-119)
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64((Float64(k * t_m) ^ 2.0) * 2.0)));
	elseif (k <= 3e+15)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) * t_m) / Float64(Float64(cos(k) * l_m) * l_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, 1.95e-119], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3e+15], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.94999999999999995e-119

   1. Initial program 57.1%

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

   2. Taylor expanded in t around 0

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

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

   4. 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. Applied rewrites 74.0%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 65.9

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

   9. Applied rewrites 65.9%

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

   if 1.94999999999999995e-119 < k < 3e15

   1. Initial program 57.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 57.1%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 62.5

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

       6. unpow-prod-down 62.5

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

       7. *-commutative 62.5

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

       8. *-commutative 62.5

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

       9. *-commutative 62.5

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

       10. unpow-prod-down 62.5

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

   11. Applied rewrites 80.3%

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

   if 3e15 < k

   1. Initial program 46.9%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   6. 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 65.8

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

   7. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      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-cos.f64 N/A

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

      6. associate-*l/ N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 68.7%

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

Alternative 14: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

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 9e-10)
    (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (* (cos k) l_m) l_m)) t_m))
    (/ 2.0 (* (/ t_m (* l_m l_m)) (* (pow (* k t_m) 2.0) 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 <= 9e-10) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / ((cos(k) * l_m) * l_m)) * t_m);
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (pow((k * t_m), 2.0) * 2.0));
	}
	return t_s * tmp;
}

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
    real(8) :: tmp
    if (t_m <= 9d-10) then
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) / ((cos(k) * l_m) * l_m)) * t_m)
    else
        tmp = 2.0d0 / ((t_m / (l_m * l_m)) * (((k * t_m) ** 2.0d0) * 2.0d0))
    end if
    code = t_s * tmp
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) {
	double tmp;
	if (t_m <= 9e-10) {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) / ((Math.cos(k) * l_m) * l_m)) * t_m);
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (Math.pow((k * t_m), 2.0) * 2.0));
	}
	return t_s * tmp;
}

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):
	tmp = 0
	if t_m <= 9e-10:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) / ((math.cos(k) * l_m) * l_m)) * t_m)
	else:
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (math.pow((k * t_m), 2.0) * 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 <= 9e-10)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(Float64(cos(k) * l_m) * l_m)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64((Float64(k * t_m) ^ 2.0) * 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-10)
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) / ((cos(k) * l_m) * l_m)) * t_m);
	else
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (((k * t_m) ^ 2.0) * 2.0));
	end
	tmp_2 = 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, 9e-10], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.9999999999999999e-10

   1. Initial program 42.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   6. 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 71.0

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

   7. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.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. associate-*r* N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 71.0

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

   9. Applied rewrites 71.0%

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

   if 8.9999999999999999e-10 < 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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.3%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 71.5

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

   9. Applied rewrites 71.5%

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

Alternative 15: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

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 9e-10)
    (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (* l_m l_m))) t_m))
    (/ 2.0 (* (/ t_m (* l_m l_m)) (* (pow (* k t_m) 2.0) 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 <= 9e-10) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (pow((k * t_m), 2.0) * 2.0));
	}
	return t_s * tmp;
}

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
    real(8) :: tmp
    if (t_m <= 9d-10) then
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) / (cos(k) * (l_m * l_m))) * t_m)
    else
        tmp = 2.0d0 / ((t_m / (l_m * l_m)) * (((k * t_m) ** 2.0d0) * 2.0d0))
    end if
    code = t_s * tmp
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) {
	double tmp;
	if (t_m <= 9e-10) {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) / (Math.cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (Math.pow((k * t_m), 2.0) * 2.0));
	}
	return t_s * tmp;
}

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):
	tmp = 0
	if t_m <= 9e-10:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) / (math.cos(k) * (l_m * l_m))) * t_m)
	else:
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (math.pow((k * t_m), 2.0) * 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 <= 9e-10)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64((Float64(k * t_m) ^ 2.0) * 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9e-10)
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	else
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (((k * t_m) ^ 2.0) * 2.0));
	end
	tmp_2 = 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, 9e-10], 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], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.9999999999999999e-10

   1. Initial program 42.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   6. 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 71.0

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

   7. Applied rewrites 71.0%

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

   if 8.9999999999999999e-10 < 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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.3%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 71.5

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

   9. Applied rewrites 71.5%

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

Alternative 16: 66.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(-t\_m\right) \cdot t\_m\\ t_3 := \mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - t\_2\right)\\ t_4 := \mathsf{fma}\left(t\_3, -0.5, 0.08333333333333333 \cdot \left(t\_m \cdot t\_m\right)\right)\\ t_5 := 0.08888888888888889 \cdot \left(t\_m \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t\_m \cdot t\_m, 0.044444444444444446\right) - \mathsf{fma}\left(\left(t\_5 - 0.3333333333333333\right) - t\_4, -0.5, \mathsf{fma}\left(t\_3, 0.041666666666666664, -0.002777777777777778 \cdot \left(t\_m \cdot t\_m\right)\right)\right), k \cdot k, t\_5\right) - \left(t\_4 + 0.3333333333333333\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot -0.6666666666666666\right) + 1\right) - t\_2, k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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_3 (fma (* -0.6666666666666666 t_m) t_m (- 1.0 t_2)))
        (t_4 (fma t_3 -0.5 (* 0.08333333333333333 (* t_m t_m))))
        (t_5 (* 0.08888888888888889 (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 2.45e-194)
      (/
       2.0
       (*
        (/ (/ t_m l_m) l_m)
        (*
         (fma
          (-
           (+
            (fma
             (-
              (fma
               (-
                (fma -0.006349206349206349 (* t_m t_m) 0.044444444444444446)
                (fma
                 (- (- t_5 0.3333333333333333) t_4)
                 -0.5
                 (fma
                  t_3
                  0.041666666666666664
                  (* -0.002777777777777778 (* t_m t_m)))))
               (* k k)
               t_5)
              (+ t_4 0.3333333333333333))
             (* k k)
             (* (* t_m t_m) -0.6666666666666666))
            1.0)
           t_2)
          (* k k)
          (* (* t_m t_m) 2.0))
         (* k k))))
      (/
       2.0
       (* (/ (* (pow (* k t_m) 2.0) 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 t_2 = -t_m * t_m;
	double t_3 = fma((-0.6666666666666666 * t_m), t_m, (1.0 - t_2));
	double t_4 = fma(t_3, -0.5, (0.08333333333333333 * (t_m * t_m)));
	double t_5 = 0.08888888888888889 * (t_m * t_m);
	double tmp;
	if (t_m <= 2.45e-194) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(((fma((fma((fma(-0.006349206349206349, (t_m * t_m), 0.044444444444444446) - fma(((t_5 - 0.3333333333333333) - t_4), -0.5, fma(t_3, 0.041666666666666664, (-0.002777777777777778 * (t_m * t_m))))), (k * k), t_5) - (t_4 + 0.3333333333333333)), (k * k), ((t_m * t_m) * -0.6666666666666666)) + 1.0) - t_2), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 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)
	t_2 = Float64(Float64(-t_m) * t_m)
	t_3 = fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - t_2))
	t_4 = fma(t_3, -0.5, Float64(0.08333333333333333 * Float64(t_m * t_m)))
	t_5 = Float64(0.08888888888888889 * Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 2.45e-194)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(Float64(Float64(fma(Float64(fma(Float64(fma(-0.006349206349206349, Float64(t_m * t_m), 0.044444444444444446) - fma(Float64(Float64(t_5 - 0.3333333333333333) - t_4), -0.5, fma(t_3, 0.041666666666666664, Float64(-0.002777777777777778 * Float64(t_m * t_m))))), Float64(k * k), t_5) - Float64(t_4 + 0.3333333333333333)), Float64(k * k), Float64(Float64(t_m * t_m) * -0.6666666666666666)) + 1.0) - t_2), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 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_] := Block[{t$95$2 = N[((-t$95$m) * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * -0.5 + N[(0.08333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.08888888888888889 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.45e-194], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.006349206349206349 * N[(t$95$m * t$95$m), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] - N[(N[(N[(t$95$5 - 0.3333333333333333), $MachinePrecision] - t$95$4), $MachinePrecision] * -0.5 + N[(t$95$3 * 0.041666666666666664 + N[(-0.002777777777777778 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + t$95$5), $MachinePrecision] - N[(t$95$4 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $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 t < 2.45000000000000002e-194

   1. Initial program 25.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 69.0%

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

   6. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(\left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right) - \left(\frac{-1}{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \left(\frac{1}{3} + \left(\frac{-1}{2} \cdot \left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) - -1 \cdot {t}^{2}\right) + \frac{1}{12} \cdot {t}^{2}\right)\right)\right) + \left(\frac{-1}{360} \cdot {t}^{2} + \frac{1}{24} \cdot \left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) - -1 \cdot {t}^{2}\right)\right)\right)\right)\right) - \left(\frac{1}{3} + \left(\frac{-1}{2} \cdot \left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) - -1 \cdot {t}^{2}\right) + \frac{1}{12} \cdot {t}^{2}\right)\right)\right)\right)\right) - -1 \cdot {t}^{2}\right)\right)}\right)} \]

   11. Applied rewrites 62.5%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right) - \mathsf{fma}\left(\left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), -0.5, 0.08333333333333333 \cdot \left(t \cdot t\right)\right), -0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), 0.041666666666666664, -0.002777777777777778 \cdot \left(t \cdot t\right)\right)\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), -0.5, 0.08333333333333333 \cdot \left(t \cdot t\right)\right) + 0.3333333333333333\right), k \cdot k, \left(t \cdot t\right) \cdot -0.6666666666666666\right) + 1\right) - \left(-t\right) \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.45000000000000002e-194 < t

   1. Initial program 61.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

Alternative 17: 67.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t\_m \cdot t\_m, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t\_m \cdot t\_m\right)\right) - 0.3333333333333333, k \cdot k, \left(t\_m \cdot t\_m\right) \cdot -0.6666666666666666\right) + 1, k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 2.45e-194)
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (/
       (*
        (fma
         (+
          (fma
           (-
            (fma
             (fma -0.006349206349206349 (* t_m t_m) 0.044444444444444446)
             (* k k)
             (* 0.08888888888888889 (* t_m t_m)))
            0.3333333333333333)
           (* k k)
           (* (* t_m t_m) -0.6666666666666666))
          1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (cos k))))
    (/ 2.0 (* (/ (* (pow (* k t_m) 2.0) 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 (t_m <= 2.45e-194) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * ((fma((fma((fma(fma(-0.006349206349206349, (t_m * t_m), 0.044444444444444446), (k * k), (0.08888888888888889 * (t_m * t_m))) - 0.3333333333333333), (k * k), ((t_m * t_m) * -0.6666666666666666)) + 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / cos(k)));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 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 (t_m <= 2.45e-194)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(Float64(fma(Float64(fma(Float64(fma(fma(-0.006349206349206349, Float64(t_m * t_m), 0.044444444444444446), Float64(k * k), Float64(0.08888888888888889 * Float64(t_m * t_m))) - 0.3333333333333333), Float64(k * k), Float64(Float64(t_m * t_m) * -0.6666666666666666)) + 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 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[t$95$m, 2.45e-194], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.006349206349206349 * N[(t$95$m * t$95$m), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(0.08888888888888889 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $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[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $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 t < 2.45000000000000002e-194

   1. Initial program 25.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 69.0%

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

   6. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos \color{blue}{k}}} \]
   10. Step-by-step derivation

       1. unpow-prod-down N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k}} \]

   11. Applied rewrites 63.9%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, \left(t \cdot t\right) \cdot -0.6666666666666666\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos \color{blue}{k}}} \]

   if 2.45000000000000002e-194 < t

   1. Initial program 61.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

Alternative 18: 66.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t\_m \cdot t\_m, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t\_m \cdot t\_m\right)\right) - 0.3333333333333333, k \cdot k, \left(t\_m \cdot t\_m\right) \cdot -0.6666666666666666\right) + 1, k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \frac{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

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 2.45e-194)
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (/
       (*
        (fma
         (+
          (fma
           (-
            (fma
             (fma -0.006349206349206349 (* t_m t_m) 0.044444444444444446)
             (* k k)
             (* 0.08888888888888889 (* t_m t_m)))
            0.3333333333333333)
           (* k k)
           (* (* t_m t_m) -0.6666666666666666))
          1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (cos k))))
    (/
     2.0
     (* (/ t_m (* l_m l_m)) (/ (* (* (* k t_m) (* k t_m)) 2.0) (cos k)))))))

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 <= 2.45e-194) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * ((fma((fma((fma(fma(-0.006349206349206349, (t_m * t_m), 0.044444444444444446), (k * k), (0.08888888888888889 * (t_m * t_m))) - 0.3333333333333333), (k * k), ((t_m * t_m) * -0.6666666666666666)) + 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / cos(k)));
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((((k * t_m) * (k * t_m)) * 2.0) / cos(k)));
	}
	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 <= 2.45e-194)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(Float64(fma(Float64(fma(Float64(fma(fma(-0.006349206349206349, Float64(t_m * t_m), 0.044444444444444446), Float64(k * k), Float64(0.08888888888888889 * Float64(t_m * t_m))) - 0.3333333333333333), Float64(k * k), Float64(Float64(t_m * t_m) * -0.6666666666666666)) + 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * 2.0) / cos(k))));
	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, 2.45e-194], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.006349206349206349 * N[(t$95$m * t$95$m), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(0.08888888888888889 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $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[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.45000000000000002e-194

   1. Initial program 25.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 69.0%

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

   6. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos \color{blue}{k}}} \]
   10. Step-by-step derivation

       1. unpow-prod-down N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       5. unpow-prod-down N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k}} \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k}} \]

   11. Applied rewrites 63.9%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, \left(t \cdot t\right) \cdot -0.6666666666666666\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos \color{blue}{k}}} \]

   if 2.45000000000000002e-194 < t

   1. Initial program 61.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.0%

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

   6. Applied rewrites 74.7%

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

   7. Taylor expanded in k around 0

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

      1. unpow-prod-down N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 67.0

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

   9. Applied rewrites 67.0%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 67.0

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

   11. Applied rewrites 67.0%

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

Alternative 19: 66.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(-t\_m\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-194}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, \left(\left(0.08888888888888889 \cdot \left(t\_m \cdot t\_m\right) - 0.3333333333333333\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - t\_2\right), -0.5, 0.08333333333333333 \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(k \cdot k\right)\right) + 1\right) - t\_2, k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \frac{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot 2}{\cos k}}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= t_m 2.45e-194)
      (/
       2.0
       (*
        (/ (/ t_m l_m) l_m)
        (*
         (fma
          (-
           (+
            (fma
             (* -0.6666666666666666 t_m)
             t_m
             (*
              (-
               (- (* 0.08888888888888889 (* t_m t_m)) 0.3333333333333333)
               (fma
                (fma (* -0.6666666666666666 t_m) t_m (- 1.0 t_2))
                -0.5
                (* 0.08333333333333333 (* t_m t_m))))
              (* k k)))
            1.0)
           t_2)
          (* k k)
          (* (* t_m t_m) 2.0))
         (* k k))))
      (/
       2.0
       (* (/ t_m (* l_m l_m)) (/ (* (* (* k t_m) (* k t_m)) 2.0) (cos k))))))))

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;
	double tmp;
	if (t_m <= 2.45e-194) {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(((fma((-0.6666666666666666 * t_m), t_m, ((((0.08888888888888889 * (t_m * t_m)) - 0.3333333333333333) - fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - t_2)), -0.5, (0.08333333333333333 * (t_m * t_m)))) * (k * k))) + 1.0) - t_2), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * ((((k * t_m) * (k * t_m)) * 2.0) / cos(k)));
	}
	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)
	tmp = 0.0
	if (t_m <= 2.45e-194)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(Float64(Float64(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(Float64(Float64(Float64(0.08888888888888889 * Float64(t_m * t_m)) - 0.3333333333333333) - fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - t_2)), -0.5, Float64(0.08333333333333333 * Float64(t_m * t_m)))) * Float64(k * k))) + 1.0) - t_2), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * 2.0) / cos(k))));
	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[((-t$95$m) * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.45e-194], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(N[(N[(N[(0.08888888888888889 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] - N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(0.08333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.45000000000000002e-194

   1. Initial program 25.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 69.0%

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

   6. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(\left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \left(\frac{1}{3} + \left(\frac{-1}{2} \cdot \left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) - -1 \cdot {t}^{2}\right) + \frac{1}{12} \cdot {t}^{2}\right)\right)\right)\right)\right) - -1 \cdot {t}^{2}\right)\right)}\right)} \]

   11. Applied rewrites 62.8%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, \left(\left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), -0.5, 0.08333333333333333 \cdot \left(t \cdot t\right)\right)\right) \cdot \left(k \cdot k\right)\right) + 1\right) - \left(-t\right) \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.45000000000000002e-194 < t

   1. Initial program 61.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 75.0%

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

   6. Applied rewrites 74.7%

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

   7. Taylor expanded in k around 0

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

      1. unpow-prod-down N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 67.0

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

   9. Applied rewrites 67.0%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 67.0

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

   11. Applied rewrites 67.0%

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

Alternative 20: 65.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 1.95e-119)
    (/ 2.0 (* (/ t_m (* l_m l_m)) (* (pow (* k t_m) 2.0) 2.0)))
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (*
       (fma
        (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
        (* k k)
        (* (* t_m t_m) 2.0))
       (* k k)))))))

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 <= 1.95e-119) {
		tmp = 2.0 / ((t_m / (l_m * l_m)) * (pow((k * t_m), 2.0) * 2.0));
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	}
	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 <= 1.95e-119)
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64((Float64(k * t_m) ^ 2.0) * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	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, 1.95e-119], N[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999995e-119

   1. Initial program 57.1%

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

   2. Taylor expanded in t around 0

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

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

   4. 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. Applied rewrites 74.0%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 65.9

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

   9. Applied rewrites 65.9%

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

   if 1.94999999999999995e-119 < k

   1. Initial program 50.1%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 72.7%

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

   6. Applied rewrites 72.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 42.0%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 43.7

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

       6. unpow-prod-down 43.7

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

       7. *-commutative 43.7

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

       8. *-commutative 43.7

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

       9. *-commutative 43.7

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

       10. unpow-prod-down 43.7

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

   11. Applied rewrites 64.1%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 64.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-143}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 7e-143)
    (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m))
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (*
       (fma
        (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
        (* k k)
        (* (* t_m t_m) 2.0))
       (* k k)))))))

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 <= 7e-143) {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	}
	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 <= 7e-143)
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	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, 7e-143], 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[(2.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.00000000000000011e-143

   1. Initial program 56.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 51.2

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

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. 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 51.2

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

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   7. 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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      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 65.1

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

   8. Applied rewrites 65.1%

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

   if 7.00000000000000011e-143 < k

   1. Initial program 50.9%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 73.0%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 42.7%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 44.4

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

       6. unpow-prod-down 44.4

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

       7. *-commutative 44.4

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

       8. *-commutative 44.4

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

       9. *-commutative 44.4

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

       10. unpow-prod-down 44.4

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

   11. Applied rewrites 64.5%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 63.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-130}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 8.5e-130)
    (* l_m (/ l_m (* k (* k (pow t_m 3.0)))))
    (/
     2.0
     (*
      (/ (/ t_m l_m) l_m)
      (*
       (fma
        (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
        (* k k)
        (* (* t_m t_m) 2.0))
       (* k k)))))))

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 <= 8.5e-130) {
		tmp = l_m * (l_m / (k * (k * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / (((t_m / l_m) / l_m) * (fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * (k * k)));
	}
	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 <= 8.5e-130)
		tmp = Float64(l_m * Float64(l_m / Float64(k * Float64(k * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k))));
	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, 8.5e-130], 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[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.50000000000000033e-130

   1. Initial program 57.1%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 51.5

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

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. 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. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      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. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 63.3

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

      16. lift-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-pow.f64 63.3

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

   6. Applied rewrites 63.3%

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

   if 8.50000000000000033e-130 < k

   1. Initial program 50.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 72.7%

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

   6. Applied rewrites 72.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 42.2%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 43.9

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

       6. unpow-prod-down 43.9

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

       7. *-commutative 43.9

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

       8. *-commutative 43.9

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

       9. *-commutative 43.9

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

       10. unpow-prod-down 43.9

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

   11. Applied rewrites 64.2%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 60.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{t\_m}{l\_m \cdot l\_m} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t\_m, t\_m, 1 - \left(-t\_m\right) \cdot t\_m\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot k\right) \cdot k\right)} \end{array} \]

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
  (/
   2.0
   (*
    (/ t_m (* l_m l_m))
    (*
     (*
      (fma
       (fma (* -0.6666666666666666 t_m) t_m (- 1.0 (* (- t_m) t_m)))
       (* k k)
       (* (* t_m t_m) 2.0))
      k)
     k)))))

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 * (2.0 / ((t_m / (l_m * l_m)) * ((fma(fma((-0.6666666666666666 * t_m), t_m, (1.0 - (-t_m * t_m))), (k * k), ((t_m * t_m) * 2.0)) * k) * k)));
}

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(2.0 / Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(Float64(fma(fma(Float64(-0.6666666666666666 * t_m), t_m, Float64(1.0 - Float64(Float64(-t_m) * t_m))), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * k) * k))))
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[(2.0 / N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.6666666666666666 * t$95$m), $MachinePrecision] * t$95$m + N[(1.0 - N[((-t$95$m) * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

2. Taylor expanded in t around 0

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

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

4. Applied rewrites 73.8%

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

6. Applied rewrites 73.6%

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

7. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 44.8%

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

11. Applied rewrites 60.2%

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

Alternative 24: 51.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)} \end{array} \]

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(Float64(l_m * 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[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 51.2

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

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
5. 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 51.2

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

6. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025096 
(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))))