Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 82.8%

Time: 9.8s

Alternatives: 26

Speedup: 6.2×

• 27.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{+122}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t\_m}{\cos k \cdot l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.75e+122)
    (/
     2.0
     (/
      (/
       (* (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)) t_m)
       (* (cos k) l_m))
      l_m))
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k)) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.75e+122) {
		tmp = 2.0 / (((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) * t_m) / (cos(k) * l_m)) / l_m);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.75e+122)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) * t_m) / Float64(cos(k) * l_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)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.75e+122], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$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] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.75000000000000007e122

   1. Initial program 52.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/r* N/A

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

   9. Applied rewrites 80.2%

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

   if 1.75000000000000007e122 < t

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 44.2

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

   6. Applied rewrites 44.2%

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

Alternative 2: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_2\right) + 1\right) \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(t\_2 + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
          (+ (+ 1.0 t_2) 1.0))
         5e+127)
      (/
       2.0
       (*
        (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k))
        (* (tan k) (+ (+ t_2 1.0) 1.0))))
      (/
       2.0
       (*
        (/
         (fma (pow (* k t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))
         (* (cos k) l_m))
        (/ t_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + t_2) + 1.0)) <= 5e+127) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l_m) / l_m) * sin(k)) * (tan(k) * ((t_2 + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / ((fma(pow((k * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / (cos(k) * l_m)) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + t_2) + 1.0)) <= 5e+127)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * Float64(tan(k) * Float64(Float64(t_2 + 1.0) + 1.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(k * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * l_m)) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[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 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5e+127], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$2 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * 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]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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))) < 5.0000000000000004e127

   1. Initial program 84.4%

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

   4. Applied rewrites 87.3%

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

   if 5.0000000000000004e127 < (*.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 29.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 70.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 69.2%

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

Alternative 3: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2 \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<=
         (* (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k)) t_2)
         5e+127)
      (/
       2.0
       (* (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k)) t_2))
      (/
       2.0
       (*
        (/
         (fma (pow (* k t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))
         (* (cos k) l_m))
        (/ t_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if (((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2) <= 5e+127) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2);
	} else {
		tmp = 2.0 / ((fma(pow((k * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / (cos(k) * l_m)) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * t_2) <= 5e+127)
		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)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(k * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * l_m)) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[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] * t$95$2), $MachinePrecision], 5e+127], 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] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * 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]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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))) < 5.0000000000000004e127

   1. Initial program 84.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 84.4

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

   4. Applied rewrites 84.4%

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

   if 5.0000000000000004e127 < (*.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 29.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 70.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 69.2%

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

Alternative 4: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{+122}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t\_m}{\cos k \cdot l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.75e+122)
    (/
     2.0
     (/
      (/
       (* (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)) t_m)
       (* (cos k) l_m))
      l_m))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      (+ (fma (/ k t_m) (/ k t_m) 1.0) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.75e+122) {
		tmp = 2.0 / (((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) * t_m) / (cos(k) * l_m)) / l_m);
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.75e+122)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) * t_m) / Float64(cos(k) * l_m)) / l_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.75e+122], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.75000000000000007e122

   1. Initial program 52.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/r* N/A

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

   9. Applied rewrites 80.2%

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

   if 1.75000000000000007e122 < t

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

Alternative 5: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sin k \cdot t\_m\right)}^{2}\\ t_3 := \cos k \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(t\_2, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t\_m}{t\_3}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_2, 2, {\sin k}^{2} \cdot \left(k \cdot k\right)\right)}{t\_3} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (* (sin k) t_m) 2.0)) (t_3 (* (cos k) l_m)))
   (*
    t_s
    (if (<= t_m 1.2e-198)
      (/ 2.0 (/ (/ (* (fma t_2 2.0 (pow (* (sin k) k) 2.0)) t_m) t_3) l_m))
      (/
       2.0
       (* (/ (fma t_2 2.0 (* (pow (sin k) 2.0) (* k k))) t_3) (/ t_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow((sin(k) * t_m), 2.0);
	double t_3 = cos(k) * l_m;
	double tmp;
	if (t_m <= 1.2e-198) {
		tmp = 2.0 / (((fma(t_2, 2.0, pow((sin(k) * k), 2.0)) * t_m) / t_3) / l_m);
	} else {
		tmp = 2.0 / ((fma(t_2, 2.0, (pow(sin(k), 2.0) * (k * k))) / t_3) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(sin(k) * t_m) ^ 2.0
	t_3 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (t_m <= 1.2e-198)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(t_2, 2.0, (Float64(sin(k) * k) ^ 2.0)) * t_m) / t_3) / l_m));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(t_2, 2.0, Float64((sin(k) ^ 2.0) * Float64(k * k))) / t_3) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-198], N[(2.0 / N[(N[(N[(N[(t$95$2 * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * 2.0 + N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.19999999999999993e-198

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/r* N/A

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

   9. Applied rewrites 82.6%

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

   if 1.19999999999999993e-198 < t

   1. Initial program 54.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 78.5%

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

       1. lift-pow.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. unpow-prod-down N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 78.5

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

   11. Applied rewrites 78.5%

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

Alternative 6: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sin k \cdot k\right)}^{2}\\ t_3 := \cos k \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(k \cdot t\_m\right)}^{2}, 2, t\_2\right)}{t\_3} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;l\_m \leq 5.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, t\_2\right) \cdot t\_m}{t\_3 \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (* (sin k) k) 2.0)) (t_3 (* (cos k) l_m)))
   (*
    t_s
    (if (<= l_m 3.8e-144)
      (/ 2.0 (* (/ (fma (pow (* k t_m) 2.0) 2.0 t_2) t_3) (/ t_m l_m)))
      (if (<= l_m 5.6e+141)
        (/ 2.0 (/ (* (fma (pow (* (sin k) t_m) 2.0) 2.0 t_2) t_m) (* t_3 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 t_2 = pow((sin(k) * k), 2.0);
	double t_3 = cos(k) * l_m;
	double tmp;
	if (l_m <= 3.8e-144) {
		tmp = 2.0 / ((fma(pow((k * t_m), 2.0), 2.0, t_2) / t_3) * (t_m / l_m));
	} else if (l_m <= 5.6e+141) {
		tmp = 2.0 / ((fma(pow((sin(k) * t_m), 2.0), 2.0, t_2) * t_m) / (t_3 * 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)
	t_2 = Float64(sin(k) * k) ^ 2.0
	t_3 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (l_m <= 3.8e-144)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(k * t_m) ^ 2.0), 2.0, t_2) / t_3) * Float64(t_m / l_m)));
	elseif (l_m <= 5.6e+141)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, t_2) * t_m) / Float64(t_3 * 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_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 3.8e-144], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 5.6e+141], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(t$95$3 * 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 3 regimes

2. if l < 3.79999999999999993e-144

   1. Initial program 55.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 80.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 76.7%

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

   if 3.79999999999999993e-144 < l < 5.59999999999999982e141

   1. Initial program 65.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 93.4%

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

   if 5.59999999999999982e141 < l

   1. Initial program 20.2%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 43.3%

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

      1. lift-*.f64 N/A

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

      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 45.9

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

   7. Applied rewrites 45.9%

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

Alternative 7: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sin k \cdot k\right)}^{2}\\ t_3 := \cos k \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 6.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(k \cdot t\_m\right)}^{2}, 2, t\_2\right)}{t\_3} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;l\_m \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, t\_2\right) \cdot \frac{t\_m}{t\_3 \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (* (sin k) k) 2.0)) (t_3 (* (cos k) l_m)))
   (*
    t_s
    (if (<= l_m 6.8e-144)
      (/ 2.0 (* (/ (fma (pow (* k t_m) 2.0) 2.0 t_2) t_3) (/ t_m l_m)))
      (if (<= l_m 2e+149)
        (/ 2.0 (* (fma (pow (* (sin k) t_m) 2.0) 2.0 t_2) (/ t_m (* t_3 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 t_2 = pow((sin(k) * k), 2.0);
	double t_3 = cos(k) * l_m;
	double tmp;
	if (l_m <= 6.8e-144) {
		tmp = 2.0 / ((fma(pow((k * t_m), 2.0), 2.0, t_2) / t_3) * (t_m / l_m));
	} else if (l_m <= 2e+149) {
		tmp = 2.0 / (fma(pow((sin(k) * t_m), 2.0), 2.0, t_2) * (t_m / (t_3 * 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)
	t_2 = Float64(sin(k) * k) ^ 2.0
	t_3 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (l_m <= 6.8e-144)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(k * t_m) ^ 2.0), 2.0, t_2) / t_3) * Float64(t_m / l_m)));
	elseif (l_m <= 2e+149)
		tmp = Float64(2.0 / Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, t_2) * Float64(t_m / Float64(t_3 * 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_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 6.8e-144], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2e+149], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision] * N[(t$95$m / N[(t$95$3 * l$95$m), $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 l < 6.80000000000000035e-144

   1. Initial program 55.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 80.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 76.7%

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

   if 6.80000000000000035e-144 < l < 2.0000000000000001e149

   1. Initial program 64.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 87.5%

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

   if 2.0000000000000001e149 < l

   1. Initial program 20.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      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 44.6

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

   7. Applied rewrites 44.6%

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

Alternative 8: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k \cdot l\_m\\ t_3 := \mathsf{fma}\left({\left(\sin k \cdot t\_m\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_3 \cdot t\_m}{t\_2}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_3}{t\_2} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (cos k) l_m))
        (t_3 (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))))
   (*
    t_s
    (if (<= t_m 1.2e-198)
      (/ 2.0 (/ (/ (* t_3 t_m) t_2) l_m))
      (/ 2.0 (* (/ t_3 t_2) (/ t_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = cos(k) * l_m;
	double t_3 = fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0));
	double tmp;
	if (t_m <= 1.2e-198) {
		tmp = 2.0 / (((t_3 * t_m) / t_2) / l_m);
	} else {
		tmp = 2.0 / ((t_3 / t_2) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(cos(k) * l_m)
	t_3 = fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.2e-198)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_3 * t_m) / t_2) / l_m));
	else
		tmp = Float64(2.0 / Float64(Float64(t_3 / t_2) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, Block[{t$95$3 = 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[(t$95$s * If[LessEqual[t$95$m, 1.2e-198], N[(2.0 / N[(N[(N[(t$95$3 * t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$3 / t$95$2), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.19999999999999993e-198

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/r* N/A

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

   9. Applied rewrites 82.6%

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

   if 1.19999999999999993e-198 < t

   1. Initial program 54.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 78.5%

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

Alternative 9: 85.1% accurate, 0.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (/
   2.0
   (*
    (/
     (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0))
     (* (cos k) l_m))
    (/ t_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (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)));
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(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))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(2.0 / N[(N[(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]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 70.3%

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

7. Applied rewrites 72.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. times-frac N/A

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

9. Applied rewrites 80.5%

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

Alternative 10: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{l\_m}}{\cos k} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.2e-25)
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m)))
    (/ 2.0 (* (/ (/ (pow (* (sin k) k) 2.0) l_m) (cos k)) (/ t_m l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.2e-25) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / l_m) / cos(k)) * (t_m / l_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 (k <= 3.2d-25) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    else
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / l_m) / cos(k)) * (t_m / l_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 (k <= 3.2e-25) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / l_m) / Math.cos(k)) * (t_m / l_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 k <= 3.2e-25:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	else:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / l_m) / math.cos(k)) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.2e-25)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / l_m) / cos(k)) * Float64(t_m / l_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 (k <= 3.2e-25)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	else
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / l_m) / cos(k)) * (t_m / l_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[k, 3.2e-25], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000001e-25

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

   7. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 68.5

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

   12. Applied rewrites 68.5%

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

   if 3.2000000000000001e-25 < k

   1. Initial program 53.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 78.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 77.4%

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

   10. Taylor expanded in t around 0

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

       1. associate-/r* N/A

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

       2. lower-/.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. lift-cos.f64 75.8

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

   12. Applied rewrites 75.8%

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

Alternative 11: 72.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 900000:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot t\_m\right)}^{2} \cdot 2}{t\_2} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2} \cdot t\_m}{t\_2 \cdot l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (cos k) l_m)))
   (*
    t_s
    (if (<= k 900000.0)
      (/ 2.0 (* (/ (* (pow (* (sin k) t_m) 2.0) 2.0) t_2) (/ t_m l_m)))
      (/ 2.0 (/ (* (pow (* (sin k) k) 2.0) t_m) (* t_2 l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = cos(k) * l_m;
	double tmp;
	if (k <= 900000.0) {
		tmp = 2.0 / (((pow((sin(k) * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) * t_m) / (t_2 * l_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) :: t_2
    real(8) :: tmp
    t_2 = cos(k) * l_m
    if (k <= 900000.0d0) then
        tmp = 2.0d0 / (((((sin(k) * t_m) ** 2.0d0) * 2.0d0) / t_2) * (t_m / l_m))
    else
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) * t_m) / (t_2 * l_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 t_2 = Math.cos(k) * l_m;
	double tmp;
	if (k <= 900000.0) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) * t_m) / (t_2 * l_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):
	t_2 = math.cos(k) * l_m
	tmp = 0
	if k <= 900000.0:
		tmp = 2.0 / (((math.pow((math.sin(k) * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m))
	else:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) * t_m) / (t_2 * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(cos(k) * l_m)
	tmp = 0.0
	if (k <= 900000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * t_m) ^ 2.0) * 2.0) / t_2) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) * t_m) / Float64(t_2 * l_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)
	t_2 = cos(k) * l_m;
	tmp = 0.0;
	if (k <= 900000.0)
		tmp = 2.0 / (((((sin(k) * t_m) ^ 2.0) * 2.0) / t_2) * (t_m / l_m));
	else
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) * t_m) / (t_2 * l_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_] := Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 900000.0], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(t$95$2 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9e5

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.6%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-pow.f64 68.8

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

   12. Applied rewrites 68.8%

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

   if 9e5 < k

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 72.9

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

   8. Applied rewrites 72.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-*l/ N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

   10. Applied rewrites 77.7%

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

Alternative 12: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2} \cdot t\_m}{\left(\cos k \cdot l\_m\right) \cdot l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-19)
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m)))
    (/ 2.0 (/ (* (pow (* (sin k) k) 2.0) t_m) (* (* (cos k) l_m) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.7e-19) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) * t_m) / ((cos(k) * l_m) * l_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 (k <= 1.7d-19) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    else
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) * t_m) / ((cos(k) * l_m) * l_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 (k <= 1.7e-19) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) * t_m) / ((Math.cos(k) * l_m) * l_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 k <= 1.7e-19:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	else:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) * t_m) / ((math.cos(k) * l_m) * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.7e-19)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) * t_m) / Float64(Float64(cos(k) * l_m) * l_m)));
	end
	return Float64(t_s * tmp)
end

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 (k <= 1.7e-19)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	else
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) * t_m) / ((cos(k) * l_m) * l_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[k, 1.7e-19], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7000000000000001e-19

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

   7. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 68.5

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

   12. Applied rewrites 68.5%

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

   if 1.7000000000000001e-19 < k

   1. Initial program 53.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 72.3

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

   8. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-*l/ N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

   10. Applied rewrites 76.9%

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

Alternative 13: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= k 4.2e-25)
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m)))
    (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (* l_m l_m))) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 4.2e-25) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((pow((sin(k) * k), 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 (k <= 4.2d-25) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    else
        tmp = 2.0d0 / ((((sin(k) * k) ** 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 (k <= 4.2e-25) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 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 k <= 4.2e-25:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	else:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 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 (k <= 4.2e-25)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 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 (k <= 4.2e-25)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	else
		tmp = 2.0 / ((((sin(k) * k) ^ 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[k, 4.2e-25], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.20000000000000005e-25

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

   7. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 68.5

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

   12. Applied rewrites 68.5%

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

   if 4.20000000000000005e-25 < k

   1. Initial program 53.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 72.3

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

   8. Applied rewrites 72.3%

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

Alternative 14: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2, -0.6666666666666666, {l\_m}^{-1}\right) + t\_2, k \cdot k, t\_2 \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (* t_m t_m) l_m)))
   (*
    t_s
    (if (<= t_m 4e-30)
      (/
       2.0
       (*
        (*
         (fma
          (+ (fma t_2 -0.6666666666666666 (pow l_m -1.0)) t_2)
          (* k k)
          (* t_2 2.0))
         (* k k))
        (/ t_m l_m)))
      (/
       2.0
       (* (/ (* (pow (* k t_m) 2.0) 2.0) (* (cos k) l_m)) (/ t_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m * t_m) / l_m;
	double tmp;
	if (t_m <= 4e-30) {
		tmp = 2.0 / ((fma((fma(t_2, -0.6666666666666666, pow(l_m, -1.0)) + t_2), (k * k), (t_2 * 2.0)) * (k * k)) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) / (cos(k) * l_m)) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m * t_m) / l_m)
	tmp = 0.0
	if (t_m <= 4e-30)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(fma(t_2, -0.6666666666666666, (l_m ^ -1.0)) + t_2), Float64(k * k), Float64(t_2 * 2.0)) * Float64(k * k)) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / Float64(cos(k) * l_m)) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-30], N[(2.0 / N[(N[(N[(N[(N[(t$95$2 * -0.6666666666666666 + N[Power[l$95$m, -1.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $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] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4e-30

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 52.2%

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

   if 4e-30 < t

   1. Initial program 62.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 64.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 79.2%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 72.4

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

   12. Applied rewrites 72.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell}, -0.6666666666666666, {\ell}^{-1}\right) + \frac{t \cdot t}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2, -0.6666666666666666, {l\_m}^{-1}\right) + t\_2, k \cdot k, t\_2 \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (* t_m t_m) l_m)))
   (*
    t_s
    (if (<= t_m 8.8e-29)
      (/
       2.0
       (*
        (*
         (fma
          (+ (fma t_2 -0.6666666666666666 (pow l_m -1.0)) t_2)
          (* k k)
          (* t_2 2.0))
         (* k k))
        (/ t_m l_m)))
      (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m * t_m) / l_m;
	double tmp;
	if (t_m <= 8.8e-29) {
		tmp = 2.0 / ((fma((fma(t_2, -0.6666666666666666, pow(l_m, -1.0)) + t_2), (k * k), (t_2 * 2.0)) * (k * k)) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m * t_m) / l_m)
	tmp = 0.0
	if (t_m <= 8.8e-29)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(fma(t_2, -0.6666666666666666, (l_m ^ -1.0)) + t_2), Float64(k * k), Float64(t_2 * 2.0)) * Float64(k * k)) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.8e-29], N[(2.0 / N[(N[(N[(N[(N[(t$95$2 * -0.6666666666666666 + N[Power[l$95$m, -1.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.79999999999999961e-29

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 74.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 52.2%

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

   if 8.79999999999999961e-29 < t

   1. Initial program 62.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 64.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 79.2%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 72.3

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

   12. Applied rewrites 72.3%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell}, -0.6666666666666666, {\ell}^{-1}\right) + \frac{t \cdot t}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-29)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
         k
         (* (* t_m t_m) 2.0))
        (* k k))
       (* (cos k) l_m))
      (/ t_m l_m)))
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 8.5e-29) {
		tmp = 2.0 / (((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / (cos(k) * l_m)) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 8.5e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(cos(k) * l_m)) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-29], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $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[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.5000000000000001e-29

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 49.1%

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

   12. Applied rewrites 54.1%

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

   if 8.5000000000000001e-29 < t

   1. Initial program 62.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 64.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 79.2%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 72.3

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

   12. Applied rewrites 72.3%

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

Alternative 17: 70.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}{\left(\cos k \cdot l\_m\right) \cdot l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-19)
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m)))
    (/ 2.0 (/ (* (* (* k k) (* k k)) t_m) (* (* (cos k) l_m) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.7e-19) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / ((cos(k) * l_m) * l_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 (k <= 1.7d-19) then
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    else
        tmp = 2.0d0 / ((((k * k) * (k * k)) * t_m) / ((cos(k) * l_m) * l_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 (k <= 1.7e-19) {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / ((Math.cos(k) * l_m) * l_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 k <= 1.7e-19:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	else:
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / ((math.cos(k) * l_m) * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.7e-19)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * Float64(k * k)) * t_m) / Float64(Float64(cos(k) * l_m) * l_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 (k <= 1.7e-19)
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	else
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / ((cos(k) * l_m) * l_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[k, 1.7e-19], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7000000000000001e-19

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.3%

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

   7. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   9. Applied rewrites 81.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 68.5

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

   12. Applied rewrites 68.5%

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

   if 1.7000000000000001e-19 < k

   1. Initial program 53.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 39.5%

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

   11. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 58.0

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

   13. Applied rewrites 58.0%

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

Alternative 18: 68.7% accurate, 3.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / (((pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_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 * (2.0d0 / (((((k * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_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 * (2.0 / (((Math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_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 * (2.0 / (((math.pow((k * t_m), 2.0) / l_m) * 2.0) * (t_m / l_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(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_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 * (2.0 / (((((k * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_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[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 70.3%

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

7. Applied rewrites 72.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. times-frac N/A

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

9. Applied rewrites 80.5%

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

10. Taylor expanded in k around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. pow-prod-down N/A

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

    5. lower-pow.f64 N/A

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

    6. lower-*.f64 64.2

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

12. Applied rewrites 64.2%

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

Alternative 19: 66.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

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 8.8e+35)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma (* t_m t_m) -0.6666666666666666 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* 1.0 (* l_m l_m)))
      t_m))
    (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 8.8e+35) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (1.0 * (l_m * l_m))) * t_m);
	} else {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 8.8e+35)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.8e+35], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.7999999999999994e35

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

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

   12. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 49.0

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

   14. Applied rewrites 49.0%

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

   if 8.7999999999999994e35 < t

   1. Initial program 59.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 64.5

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

   9. Applied rewrites 64.5%

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

Alternative 20: 62.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k \cdot k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.8e+35)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma (* t_m t_m) -0.6666666666666666 1.0)
         (* k k)
         (* (* t_m t_m) 2.0))
        (* k k))
       (* 1.0 (* l_m l_m)))
      t_m))
    (/ (* l_m l_m) (* k (* k (pow t_m 3.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 8.8e+35) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k * k), ((t_m * t_m) * 2.0)) * (k * k)) / (1.0 * (l_m * l_m))) * t_m);
	} else {
		tmp = (l_m * l_m) / (k * (k * pow(t_m, 3.0)));
	}
	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 <= 8.8e+35)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k * k), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64(k * Float64(k * (t_m ^ 3.0))));
	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, 8.8e+35], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.7999999999999994e35

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

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

   12. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 49.0

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

   14. Applied rewrites 49.0%

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

   if 8.7999999999999994e35 < t

   1. Initial program 59.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 47.6

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

   5. Applied rewrites 47.6%

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

      1. lift-*.f64 N/A

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

      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 60.9

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

   7. Applied rewrites 60.9%

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

Alternative 21: 57.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+34}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot {t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.02e+34)
    (* l_m (/ l_m (* (* k k) (pow t_m 3.0))))
    (/
     2.0
     (*
      (/
       (* (* (fma -0.3333333333333333 (* k k) 1.0) (* k k)) (* k k))
       (* 1.0 (* l_m l_m)))
      t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.02e+34) {
		tmp = l_m * (l_m / ((k * k) * pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k * k), 1.0) * (k * k)) * (k * k)) / (1.0 * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.02e+34)
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * k) * (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * Float64(k * k)) * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.02e+34], 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[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.02e34

   1. Initial program 54.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 48.9

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. 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 55.1

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 55.1%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   if 1.02e34 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 39.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 39.4%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(1 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(1 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{-1}{3} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, k \cdot k, 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, k \cdot k, 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. lift-*.f64 54.8

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 54.8%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 59.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(t\_m \cdot t\_m\right) \cdot 2\\ t_3 := 1 \cdot \left(l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k \cdot k, t\_2\right) \cdot \left(k \cdot k\right)}{t\_3} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(k \cdot k\right)}{t\_3} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (* t_m t_m) 2.0)) (t_3 (* 1.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 1.5e+36)
      (/
       2.0
       (*
        (/
         (*
          (fma (fma (* t_m t_m) -0.6666666666666666 1.0) (* k k) t_2)
          (* k k))
         t_3)
        t_m))
      (/ 2.0 (* (/ (* t_2 (* k k)) t_3) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m * t_m) * 2.0;
	double t_3 = 1.0 * (l_m * l_m);
	double tmp;
	if (t_m <= 1.5e+36) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k * k), t_2) * (k * k)) / t_3) * t_m);
	} else {
		tmp = 2.0 / (((t_2 * (k * k)) / t_3) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m * t_m) * 2.0)
	t_3 = Float64(1.0 * Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 1.5e+36)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k * k), t_2) * Float64(k * k)) / t_3) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(k * k)) / t_3) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e+36], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e36

   1. Initial program 52.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 71.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 46.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.1%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + {t}^{2} \cdot \frac{-2}{3}, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(1 + \left(t \cdot t\right) \cdot \frac{-2}{3}, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{-2}{3} + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lift-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{-2}{3}, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lift-*.f64 49.0

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 49.0%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.5e36 < t

   1. Initial program 59.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 64.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 21.2%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.2%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lift-*.f64 49.5

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 49.5%

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 59.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(t\_m \cdot t\_m\right) \cdot 2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k \cdot k, t\_2\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}{l\_m \cdot l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (* t_m t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 2.3e+17)
      (/
       2.0
       (/
        (*
         (*
          (fma (fma (* t_m t_m) -0.6666666666666666 1.0) (* k k) t_2)
          (* k k))
         t_m)
        (* l_m l_m)))
      (/ 2.0 (* (/ (* t_2 (* k k)) (* 1.0 (* l_m l_m))) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m * t_m) * 2.0;
	double tmp;
	if (t_m <= 2.3e+17) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k * k), t_2) * (k * k)) * t_m) / (l_m * l_m));
	} else {
		tmp = 2.0 / (((t_2 * (k * k)) / (1.0 * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m * t_m) * 2.0)
	tmp = 0.0
	if (t_m <= 2.3e+17)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k * k), t_2) * Float64(k * k)) * t_m) / Float64(l_m * l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.3e+17], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e17

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 72.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   7. Applied rewrites 74.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   10. Applied rewrites 50.2%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{-2}{3}, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}} \]
   12. Step-by-step derivation

   13. Applied rewrites 48.7%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}} \]

   if 2.3e17 < t

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 62.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 22.9%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 22.9%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lift-*.f64 48.8

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 48.8%

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 56.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 \cdot \left(l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+34}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{t\_2} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{t\_2} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* 1.0 (* l_m l_m))))
   (*
    t_s
    (if (<= k 1.02e+34)
      (/ 2.0 (* (/ (* (* (* t_m t_m) 2.0) (* k k)) t_2) t_m))
      (/
       2.0
       (*
        (/ (* (* (fma -0.3333333333333333 (* k k) 1.0) (* k k)) (* k k)) t_2)
        t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 1.0 * (l_m * l_m);
	double tmp;
	if (k <= 1.02e+34) {
		tmp = 2.0 / (((((t_m * t_m) * 2.0) * (k * k)) / t_2) * t_m);
	} else {
		tmp = 2.0 / ((((fma(-0.3333333333333333, (k * k), 1.0) * (k * k)) * (k * k)) / t_2) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(1.0 * Float64(l_m * l_m))
	tmp = 0.0
	if (k <= 1.02e+34)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * 2.0) * Float64(k * k)) / t_2) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(k * k), 1.0) * Float64(k * k)) * Float64(k * k)) / t_2) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.02e+34], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.02e34

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 70.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 41.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 42.3%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lift-*.f64 52.0

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 52.0%

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.02e34 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 39.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 39.4%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {k}^{2}\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(1 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(1 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(\frac{-1}{3} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, {k}^{2}, 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, k \cdot k, 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, k \cdot k, 1\right) \cdot {k}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-1}{3}, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. lift-*.f64 54.8

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 54.8%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 54.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\left(\left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (/ 2.0 (* (/ (* (* (* t_m t_m) 2.0) (* k k)) (* 1.0 (* 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) {
	return t_s * (2.0 / (((((t_m * t_m) * 2.0) * (k * k)) / (1.0 * (l_m * l_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 * (2.0d0 / (((((t_m * t_m) * 2.0d0) * (k * k)) / (1.0d0 * (l_m * l_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 * (2.0 / (((((t_m * t_m) * 2.0) * (k * k)) / (1.0 * (l_m * l_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 * (2.0 / (((((t_m * t_m) * 2.0) * (k * k)) / (1.0 * (l_m * l_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(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * 2.0) * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_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 * (2.0 / (((((t_m * t_m) * 2.0) * (k * k)) / (1.0 * (l_m * l_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[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

5. Applied rewrites 70.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

8. Applied rewrites 41.3%

   \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

9. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right) + \left(\frac{4}{45} \cdot \left(t \cdot t\right) - \frac{1}{3}\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
10. Step-by-step derivation

11. Applied rewrites 41.7%

    \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + \left(0.08888888888888889 \cdot \left(t \cdot t\right) - 0.3333333333333333\right) \cdot \left(k \cdot k\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

12. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

    2. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

    4. lift-*.f64 51.3

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

14. Applied rewrites 51.3%

    \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
15. Add Preprocessing

Alternative 26: 51.3% 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 53.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 48.1

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 48.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. unpow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lower-*.f64 48.1

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

7. Applied rewrites 48.1%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(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))))