Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 85.1%

Time: 7.7s

Alternatives: 23

Speedup: 1.3×

• 27.7% 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: 54.5% 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: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.8 \cdot 10^{+242}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 1.8e+242)
    (/
     2.0
     (*
      (/
       (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))
       (* (cos k) l_m))
      (/ t_m l_m)))
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (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 <= 1.8e+242) {
		tmp = 2.0 / ((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / (cos(k) * l_m)) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * 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 <= 1.8e+242)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * l_m)) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * 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, 1.8e+242], 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[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 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 < 1.79999999999999997e242

   1. Initial program 56.7%

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

   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 76.4%

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

   6. Applied rewrites 76.1%

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

      1. lift-/.f64 N/A

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

   8. Applied rewrites 86.0%

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

   if 1.79999999999999997e242 < l

   1. Initial program 36.9%

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

   2. Taylor expanded in t around 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 53.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 77.9

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

   6. Applied rewrites 77.9%

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

Alternative 2: 67.3% accurate, 0.6× speedup

Math

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

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 (<=
       (/
        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) (* (cos k) (* l_m l_m))) t_m))
    (/
     2.0
     (* (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (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 ((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) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k)) * tan(k)) * 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 ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 5d-219) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) / (cos(k) * (l_m * l_m))) * t_m)
    else
        tmp = 2.0d0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k)) * tan(k)) * 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 ((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))) <= 5e-219) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) / (Math.cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * Math.sin(k)) * Math.tan(k)) * 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 (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))) <= 5e-219:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) / (math.cos(k) * (l_m * l_m))) * t_m)
	else:
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * math.sin(k)) * math.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 (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(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k)) * tan(k)) * 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 ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 5e-219)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	else
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k)) * tan(k)) * 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[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[(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[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $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 (/.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)))) < 5.0000000000000002e-219

   1. Initial program 75.8%

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

   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 86.6%

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

   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 81.7

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

   7. Applied rewrites 81.7%

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

   if 5.0000000000000002e-219 < (/.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 38.9%

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

   2. Taylor expanded in t around 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 46.7%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 56.8

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

   6. Applied rewrites 56.8%

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

Alternative 3: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+287)
    (/ 2.0 (* (/ (* (pow (* k t_m) 2.0) 2.0) (* (cos k) (* l_m l_m))) t_m))
    (/ 2.0 (* (* (* k k) (/ (/ t_m l_m) l_m)) (* 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))) <= 2e+287) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2d+287) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) / (cos(k) * (l_m * l_m))) * t_m)
    else
        tmp = 2.0d0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+287) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) / (Math.cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+287:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) / (math.cos(k) * (l_m * l_m))) * t_m)
	else:
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+287)
		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));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m)) * 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))) <= 2e+287)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	else
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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], 2e+287], 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], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000002e287

   1. Initial program 77.6%

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

   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 86.5%

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

   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 79.1

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

   7. Applied rewrites 79.1%

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

   if 2.0000000000000002e287 < (/.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 34.9%

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

   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 63.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 56.0

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

   10. Applied rewrites 56.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 62.5

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

   12. Applied rewrites 62.5%

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

Alternative 4: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m))
    (/ 2.0 (* (* (* k k) (/ (/ t_m l_m) l_m)) (* 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))) <= 2e+271) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2d+271) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / (l_m * l_m)) * 2.0d0) * t_m)
    else
        tmp = 2.0d0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m)
	else:
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271)
		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(k * k) * Float64(Float64(t_m / l_m) / l_m)) * 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))) <= 2e+271)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / (l_m * l_m)) * 2.0) * t_m);
	else
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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], 2e+271], 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[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   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 86.6%

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

   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 78.2

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

   7. Applied rewrites 78.2%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 62.4

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

   12. Applied rewrites 62.4%

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

Alternative 5: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m))
    (/ 2.0 (* (* (* k k) (/ (/ t_m l_m) l_m)) (* 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))) <= 2e+271) {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2d+271) then
        tmp = (l_m * l_m) / (((k * t_m) ** 2.0d0) * t_m)
    else
        tmp = 2.0d0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271) {
		tmp = (l_m * l_m) / (Math.pow((k * t_m), 2.0) * t_m);
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271:
		tmp = (l_m * l_m) / (math.pow((k * t_m), 2.0) * t_m)
	else:
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271)
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m)) * 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))) <= 2e+271)
		tmp = (l_m * l_m) / (((k * t_m) ^ 2.0) * t_m);
	else
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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], 2e+271], 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[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. pow3 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. 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 62.6

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

   6. Applied rewrites 62.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   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. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 62.4

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

   12. Applied rewrites 62.4%

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

Alternative 6: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (/ (* l_m l_m) (* k (* k (pow t_m 3.0))))
    (/ 2.0 (* (* (* k k) (/ (/ t_m l_m) l_m)) (* 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))) <= 2e+271) {
		tmp = (l_m * l_m) / (k * (k * pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2d+271) then
        tmp = (l_m * l_m) / (k * (k * (t_m ** 3.0d0)))
    else
        tmp = 2.0d0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271) {
		tmp = (l_m * l_m) / (k * (k * Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271:
		tmp = (l_m * l_m) / (k * (k * math.pow(t_m, 3.0)))
	else:
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271)
		tmp = Float64(Float64(l_m * l_m) / Float64(k * Float64(k * (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m)) * 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))) <= 2e+271)
		tmp = (l_m * l_m) / (k * (k * (t_m ^ 3.0)));
	else
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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], 2e+271], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 70.2

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

   6. Applied rewrites 70.2%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 62.4

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

   12. Applied rewrites 62.4%

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

Alternative 7: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot {t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (* l_m (/ l_m (* (* k k) (pow t_m 3.0))))
    (/ 2.0 (* (* (* k k) (/ (/ t_m l_m) l_m)) (* 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))) <= 2e+271) {
		tmp = l_m * (l_m / ((k * k) * pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2d+271) then
        tmp = l_m * (l_m / ((k * k) * (t_m ** 3.0d0)))
    else
        tmp = 2.0d0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271) {
		tmp = l_m * (l_m / ((k * k) * Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271:
		tmp = l_m * (l_m / ((k * k) * math.pow(t_m, 3.0)))
	else:
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271)
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * k) * (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m)) * 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))) <= 2e+271)
		tmp = l_m * (l_m / ((k * k) * (t_m ^ 3.0)));
	else
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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], 2e+271], N[(l$95$m * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 63.7

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

   6. Applied rewrites 63.7%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 62.4

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

   12. Applied rewrites 62.4%

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

Alternative 8: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(\sin k \cdot t\_m\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 l_m) 0.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) k)
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= (* l_m l_m) 2e+293)
      (/
       2.0
       (*
        (/
         (fma (pow (sin k) 2.0) (* k k) (* (pow (* (sin k) t_m) 2.0) 2.0))
         (* (cos k) (* l_m l_m)))
        t_m))
      (/
       2.0
       (*
        (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (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 * l_m) <= 0.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if ((l_m * l_m) <= 2e+293) {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (pow((sin(k) * t_m), 2.0) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * 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 (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (Float64(l_m * l_m) <= 2e+293)
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64((Float64(sin(k) * t_m) ^ 2.0) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * 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[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], 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] * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+293], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 83.0

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

   3. Applied rewrites 83.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 82.9%

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

   if 0.0 < (*.f64 l l) < 1.9999999999999998e293

   1. Initial program 64.2%

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

   2. Taylor expanded in t around 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 87.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 87.8%

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

   if 1.9999999999999998e293 < (*.f64 l l)

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 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 49.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 71.6

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

   6. Applied rewrites 71.6%

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

Alternative 9: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t\_m}{\left(l\_m \cdot l\_m\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 l_m) 0.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) k)
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= (* l_m l_m) 2e+293)
      (/
       2.0
       (/
        (* (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)) t_m)
        (* (* l_m l_m) (cos k))))
      (/
       2.0
       (*
        (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (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 * l_m) <= 0.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if ((l_m * l_m) <= 2e+293) {
		tmp = 2.0 / ((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) * t_m) / ((l_m * l_m) * cos(k)));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * 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 (Float64(l_m * l_m) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (Float64(l_m * l_m) <= 2e+293)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) * t_m) / Float64(Float64(l_m * l_m) * cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * 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[N[(l$95$m * l$95$m), $MachinePrecision], 0.0], 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] * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 2e+293], 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] * t$95$m), $MachinePrecision] / N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 83.0

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

   3. Applied rewrites 83.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 82.9%

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

   if 0.0 < (*.f64 l l) < 1.9999999999999998e293

   1. Initial program 64.2%

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

   2. Taylor expanded in t around 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 87.8%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if 1.9999999999999998e293 < (*.f64 l l)

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 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 49.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 71.6

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

   6. Applied rewrites 71.6%

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

Alternative 10: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t\_m}{\left(l\_m \cdot l\_m\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 l_m) 2e+293)
    (/
     2.0
     (/
      (* (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)) t_m)
      (* (* l_m l_m) (cos k))))
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (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 * l_m) <= 2e+293) {
		tmp = 2.0 / ((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) * t_m) / ((l_m * l_m) * cos(k)));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * 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 (Float64(l_m * l_m) <= 2e+293)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) * t_m) / Float64(Float64(l_m * l_m) * cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * 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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+293], 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] * t$95$m), $MachinePrecision] / N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.9999999999999998e293

   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 82.0%

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

   6. Applied rewrites 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if 1.9999999999999998e293 < (*.f64 l l)

   1. Initial program 35.5%

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

   2. Taylor expanded in t around 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 49.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 71.6

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

   6. Applied rewrites 71.6%

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

Alternative 11: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 1.05e+148)
    (/
     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 (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (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 <= 1.05e+148) {
		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(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * 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 <= 1.05e+148)
		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(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * 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, 1.05e+148], 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[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $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 < 1.04999999999999999e148

   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 82.0%

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

   if 1.04999999999999999e148 < l

   1. Initial program 35.3%

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

   2. Taylor expanded in t around 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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 71.7

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

   6. Applied rewrites 71.7%

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

Alternative 12: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 1.05e+148)
    (/
     2.0
     (*
      (/
       (fma (pow (sin k) 2.0) (* k k) (* (pow (* k t_m) 2.0) 2.0))
       (* (cos k) (* l_m l_m)))
      t_m))
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (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 <= 1.05e+148) {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (pow((k * t_m), 2.0) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * 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 <= 1.05e+148)
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64((Float64(k * t_m) ^ 2.0) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * 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, 1.05e+148], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.04999999999999999e148

   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 82.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 82.0%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 79.9%

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

   if 1.04999999999999999e148 < l

   1. Initial program 35.3%

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

   2. Taylor expanded in t around 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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 71.7

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

   6. Applied rewrites 71.7%

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

Alternative 13: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log l\_m, -2, \log t\_m \cdot 3\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 1.05e+148)
    (/
     2.0
     (*
      (/
       (fma (pow (sin k) 2.0) (* k k) (* (pow (* k t_m) 2.0) 2.0))
       (* (cos k) (* l_m l_m)))
      t_m))
    (/
     2.0
     (*
      (* (* (exp (fma (log l_m) -2.0 (* (log t_m) 3.0))) (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 <= 1.05e+148) {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (pow((k * t_m), 2.0) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(l_m), -2.0, (log(t_m) * 3.0))) * 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 <= 1.05e+148)
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64((Float64(k * t_m) ^ 2.0) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(l_m), -2.0, Float64(log(t_m) * 3.0))) * 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, 1.05e+148], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[l$95$m], $MachinePrecision] * -2.0 + N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $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 < 1.04999999999999999e148

   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 82.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 82.0%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 79.9%

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

   if 1.04999999999999999e148 < l

   1. Initial program 35.3%

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

   2. Taylor expanded in t around 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. exp-diff 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. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 71.7

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

   6. Applied rewrites 71.7%

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

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 71.7

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

   8. Applied rewrites 71.7%

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

Alternative 14: 62.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (* k k) (/ (/ t_m l_m) l_m)) (* 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))) <= 2e+271) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2d+271) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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))) <= 2e+271)
		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(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m)) * 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))) <= 2e+271)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((k * k) * ((t_m / l_m) / l_m)) * (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], 2e+271], 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[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. pow3 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. 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 62.6

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

   6. Applied rewrites 62.6%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 62.4

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

   12. Applied rewrites 62.4%

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

Alternative 15: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{l\_m \cdot l\_m} \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+271)
    (/ (* 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))) <= 2e+271) {
		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))) <= 2d+271) 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))) <= 2e+271) {
		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))) <= 2e+271:
		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))) <= 2e+271)
		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(Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(k * k)) * 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))) <= 2e+271)
		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], 2e+271], 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[(N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. pow3 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. 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 62.6

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

   6. Applied rewrites 62.6%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   12. Applied rewrites 55.9%

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

Alternative 16: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (* k k) (/ t_m (* l_m l_m))) (* 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))) <= 2e+271) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (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))) <= 2d+271) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((k * k) * (t_m / (l_m * l_m))) * (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))) <= 2e+271) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (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))) <= 2e+271:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (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))) <= 2e+271)
		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(Float64(k * k) * Float64(t_m / Float64(l_m * l_m))) * 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))) <= 2e+271)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((k * k) * (t_m / (l_m * l_m))) * (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], 2e+271], 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[(N[(k * k), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. pow3 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. 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 62.6

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

   6. Applied rewrites 62.6%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

Alternative 17: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot \frac{t\_m}{l\_m \cdot l\_m}\right)\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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)))
       2e+271)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* k (* k (/ t_m (* l_m l_m)))) (* 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))) <= 2e+271) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((k * (k * (t_m / (l_m * l_m)))) * (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))) <= 2d+271) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / ((k * (k * (t_m / (l_m * l_m)))) * (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))) <= 2e+271) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((k * (k * (t_m / (l_m * l_m)))) * (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))) <= 2e+271:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / ((k * (k * (t_m / (l_m * l_m)))) * (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))) <= 2e+271)
		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(k * Float64(k * Float64(t_m / Float64(l_m * l_m)))) * 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))) <= 2e+271)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / ((k * (k * (t_m / (l_m * l_m)))) * (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], 2e+271], 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[(k * N[(k * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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)))) < 1.99999999999999991e271

   1. Initial program 77.6%

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

   2. Taylor expanded in 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 62.6

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

   4. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. pow3 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. 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 62.6

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

   6. Applied rewrites 62.6%

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

   if 1.99999999999999991e271 < (/.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 35.0%

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

   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 63.7%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 55.9

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

   10. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 55.9

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

   12. Applied rewrites 55.9%

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

Alternative 18: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 9.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\sin k}^{2}, k \cdot k, {\left(k \cdot t\_m\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 9.2e+147)
    (/
     2.0
     (*
      (/
       (fma (pow (sin k) 2.0) (* k k) (* (pow (* k t_m) 2.0) 2.0))
       (* (cos k) (* l_m l_m)))
      t_m))
    (/
     2.0
     (* (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (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 <= 9.2e+147) {
		tmp = 2.0 / ((fma(pow(sin(k), 2.0), (k * k), (pow((k * t_m), 2.0) * 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * 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 <= 9.2e+147)
		tmp = Float64(2.0 / Float64(Float64(fma((sin(k) ^ 2.0), Float64(k * k), Float64((Float64(k * t_m) ^ 2.0) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * 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, 9.2e+147], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $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 < 9.1999999999999997e147

   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 82.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 82.0%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 79.9%

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

   if 9.1999999999999997e147 < l

   1. Initial program 35.3%

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

   2. Taylor expanded in t around 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. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 19: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 860000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot l\_m}}{l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= t_m 860000000000.0)
    (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) (* (cos k) l_m)) l_m) t_m))
    (/
     2.0
     (* (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k)) k) 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 860000000000.0) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / (cos(k) * l_m)) / l_m) * t_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k)) * k) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 860000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * l_m)) / l_m) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k)) * k) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 860000000000.0], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $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[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.6e11

   1. Initial program 44.6%

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

   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 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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 71.7%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 70.6

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

   9. Applied rewrites 70.6%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sin.f64 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-*.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-cos.f64 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-pow.f64 N/A

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

       14. lift-cos.f64 N/A

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

       15. lift-*.f64 77.3

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

   11. Applied rewrites 77.3%

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

   if 8.6e11 < t

   1. Initial program 65.6%

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

   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 61.6%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. exp-diff N/A

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

      8. lower-exp.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 73.0

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

   9. Applied rewrites 73.0%

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

Alternative 20: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot l\_m}}{l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

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 6.2e-6)
    (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) (* (cos k) l_m)) l_m) t_m))
    (/ 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 <= 6.2e-6) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / (cos(k) * l_m)) / l_m) * t_m);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	}
	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 <= 6.2d-6) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / (cos(k) * l_m)) / l_m) * t_m)
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) / (cos(k) * (l_m * l_m))) * t_m)
    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 <= 6.2e-6) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / (Math.cos(k) * l_m)) / l_m) * t_m);
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) / (Math.cos(k) * (l_m * l_m))) * t_m);
	}
	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 <= 6.2e-6:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / (math.cos(k) * l_m)) / l_m) * t_m)
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) / (math.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 <= 6.2e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * l_m)) / l_m) * t_m));
	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

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 <= 6.2e-6)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / (cos(k) * l_m)) / l_m) * t_m);
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	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, 6.2e-6], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $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[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 < 6.1999999999999999e-6

   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 71.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 71.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 70.8

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

   9. Applied rewrites 70.8%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sin.f64 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-*.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-cos.f64 N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-pow.f64 N/A

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

       14. lift-cos.f64 N/A

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

       15. lift-*.f64 77.7

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

   11. Applied rewrites 77.7%

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

   if 6.1999999999999999e-6 < t

   1. Initial program 66.4%

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

   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 77.0%

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

   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 73.7

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

   7. Applied rewrites 73.7%

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

Alternative 21: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2} \cdot t\_m}{\left(\cos k \cdot l\_m\right) \cdot l\_m}}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

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 1.25e-57)
    (/ 2.0 (/ (* (pow (* (sin k) k) 2.0) t_m) (* (* (cos k) l_m) l_m)))
    (if (<= t_m 5.2e+81)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k))
        (+ (fma (/ k t_m) (/ k t_m) 1.0) 1.0)))
      (/
       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 <= 1.25e-57) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) * t_m) / ((cos(k) * l_m) * l_m));
	} else if (t_m <= 5.2e+81) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) + 1.0));
	} 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 <= 1.25e-57)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) * t_m) / Float64(Float64(cos(k) * l_m) * l_m)));
	elseif (t_m <= 5.2e+81)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) + 1.0)));
	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, 1.25e-57], 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], If[LessEqual[t$95$m, 5.2e+81], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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 3 regimes

2. if t < 1.25e-57

   1. Initial program 36.7%

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

   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 71.6%

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

   6. Applied rewrites 72.8%

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

   7. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 72.8

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

   9. Applied rewrites 72.8%

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

   if 1.25e-57 < t < 5.19999999999999984e81

   1. Initial program 74.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.3

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

   3. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if 5.19999999999999984e81 < t

   1. Initial program 62.5%

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

   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 77.4%

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

   5. Taylor expanded in 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 75.6

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

   7. Applied rewrites 75.6%

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

Alternative 22: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

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 5.5e-58)
    (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (* (cos k) l_m) l_m)) t_m))
    (if (<= t_m 5.2e+81)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k))
        (+ (fma (/ k t_m) (/ k t_m) 1.0) 1.0)))
      (/
       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 <= 5.5e-58) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / ((cos(k) * l_m) * l_m)) * t_m);
	} else if (t_m <= 5.2e+81) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) + 1.0));
	} 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 <= 5.5e-58)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(Float64(cos(k) * l_m) * l_m)) * t_m));
	elseif (t_m <= 5.2e+81)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) + 1.0)));
	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, 5.5e-58], 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], If[LessEqual[t$95$m, 5.2e+81], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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 3 regimes

2. if t < 5.49999999999999996e-58

   1. Initial program 36.7%

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

   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 71.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 71.6

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

   9. Applied rewrites 71.6%

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

   if 5.49999999999999996e-58 < t < 5.19999999999999984e81

   1. Initial program 74.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.2

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

   3. Applied rewrites 74.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if 5.19999999999999984e81 < t

   1. Initial program 62.5%

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

   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 77.4%

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

   5. Taylor expanded in 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 75.6

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

   7. Applied rewrites 75.6%

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

Alternative 23: 50.9% accurate, 12.5× speedup

Math

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

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 54.5%

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

2. Taylor expanded in 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 50.9

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

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
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. pow3 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. 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 50.9

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

6. Applied rewrites 50.9%

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

Reproduce

herbie shell --seed 2025098 
(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))))