Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 84.8%

Time: 8.0s

Alternatives: 14

Speedup: 6.6×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% 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: 84.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{t\_m \cdot t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-91}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\right)\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+45}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot t\_m\right) \cdot \left(\sin k \cdot \left(\frac{\tan k}{\ell} \cdot \mathsf{fma}\left(k, t\_2, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\ell}{\left(\sin k \cdot t\_m\right) \cdot t\_m}}{\left(\mathsf{fma}\left(t\_2, k, 2\right) \cdot \tan k\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ k (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 2.2e-91)
      (* l (* 2.0 (/ (* l (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 2.55e+45)
        (/
         2.0
         (*
          (* (* (/ t_m l) t_m) t_m)
          (* (sin k) (* (/ (tan k) l) (fma k t_2 2.0)))))
        (*
         l
         (/
          (* 2.0 (/ l (* (* (sin k) t_m) t_m)))
          (* (* (fma t_2 k 2.0) (tan k)) t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = k / (t_m * t_m);
	double tmp;
	if (t_m <= 2.2e-91) {
		tmp = l * (2.0 * ((l * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 2.55e+45) {
		tmp = 2.0 / ((((t_m / l) * t_m) * t_m) * (sin(k) * ((tan(k) / l) * fma(k, t_2, 2.0))));
	} else {
		tmp = l * ((2.0 * (l / ((sin(k) * t_m) * t_m))) / ((fma(t_2, k, 2.0) * tan(k)) * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 2.2e-91)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 2.55e+45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * t_m) * Float64(sin(k) * Float64(Float64(tan(k) / l) * fma(k, t_2, 2.0)))));
	else
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l / Float64(Float64(sin(k) * t_m) * t_m))) / Float64(Float64(fma(t_2, k, 2.0) * tan(k)) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := Block[{t$95$2 = N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.2e-91], N[(l * N[(2.0 * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+45], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 * N[(l / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$2 * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2000000000000001e-91

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if 2.2000000000000001e-91 < t < 2.5499999999999999e45

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.5%

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

   if 2.5499999999999999e45 < t

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 63.5

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

   7. Applied rewrites 63.5%

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

Alternative 2: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\\ t_3 := \sin k \cdot t\_m\\ t_4 := \frac{\ell}{t\_3}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_4 \cdot \frac{\frac{\ell}{k}}{t\_m}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+125}:\\ \;\;\;\;t\_4 \cdot \frac{\frac{\ell + \ell}{t\_2}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\ell}{t\_3 \cdot t\_m}}{t\_2 \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)))
        (t_3 (* (sin k) t_m))
        (t_4 (/ l t_3)))
   (*
    t_s
    (if (<= t_m 1.05e-161)
      (/ (* t_4 (/ (/ l k) t_m)) t_m)
      (if (<= t_m 1.2e+125)
        (* t_4 (/ (/ (+ l l) t_2) (* t_m t_m)))
        (* l (/ (* 2.0 (/ l (* t_3 t_m))) (* t_2 t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma((k / (t_m * t_m)), k, 2.0) * tan(k);
	double t_3 = sin(k) * t_m;
	double t_4 = l / t_3;
	double tmp;
	if (t_m <= 1.05e-161) {
		tmp = (t_4 * ((l / k) / t_m)) / t_m;
	} else if (t_m <= 1.2e+125) {
		tmp = t_4 * (((l + l) / t_2) / (t_m * t_m));
	} else {
		tmp = l * ((2.0 * (l / (t_3 * t_m))) / (t_2 * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k))
	t_3 = Float64(sin(k) * t_m)
	t_4 = Float64(l / t_3)
	tmp = 0.0
	if (t_m <= 1.05e-161)
		tmp = Float64(Float64(t_4 * Float64(Float64(l / k) / t_m)) / t_m);
	elseif (t_m <= 1.2e+125)
		tmp = Float64(t_4 * Float64(Float64(Float64(l + l) / t_2) / Float64(t_m * t_m)));
	else
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l / Float64(t_3 * t_m))) / Float64(t_2 * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := Block[{t$95$2 = N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(l / t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-161], N[(N[(t$95$4 * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+125], N[(t$95$4 * N[(N[(N[(l + l), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 * N[(l / N[(t$95$3 * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.05e-161

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}}}{t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\frac{\ell}{k}}{t}}{t} \]

      8. lower-/.f64 68.9

         \[\leadsto \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k}}{t}}}{t} \]

   10. Applied rewrites 68.9%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}}}{t} \]

   if 1.05e-161 < t < 1.2e125

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

   7. Applied rewrites 61.7%

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

   if 1.2e125 < t

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 63.5

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

   7. Applied rewrites 63.5%

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

Alternative 3: 75.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \tan k} \cdot \left(\frac{\frac{\ell}{k \cdot t\_m}}{t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\_m\right)} \cdot \frac{\ell}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.75e-6)
    (*
     (/ 2.0 (* (fma (/ k t_m) (/ k t_m) 2.0) (tan k)))
     (* (/ (/ l (* k t_m)) t_m) (/ l t_m)))
    (/
     (*
      (/ (+ l l) (* (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) (* (sin k) t_m)))
      (/ l t_m))
     t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e-6) {
		tmp = (2.0 / (fma((k / t_m), (k / t_m), 2.0) * tan(k))) * (((l / (k * t_m)) / t_m) * (l / t_m));
	} else {
		tmp = (((l + l) / ((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) * (sin(k) * t_m))) * (l / t_m)) / t_m;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.75e-6)
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * tan(k))) * Float64(Float64(Float64(l / Float64(k * t_m)) / t_m) * Float64(l / t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) * Float64(sin(k) * t_m))) * Float64(l / t_m)) / t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 1.75e-6], N[(N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.74999999999999997e-6

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 1.74999999999999997e-6 < k

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 65.9%

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

Alternative 4: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \tan k} \cdot \left(\frac{\frac{\ell}{k \cdot t\_m}}{t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell + \ell}{\left(\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\right) \cdot t\_m\right) \cdot \sin k}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot t\_m\right) \cdot t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.7e-58)
    (*
     (/ 2.0 (* (fma (/ k t_m) (/ k t_m) 2.0) (tan k)))
     (* (/ (/ l (* k t_m)) t_m) (/ l t_m)))
    (if (<= k 1.7e+127)
      (*
       l
       (/
        (/
         (+ l l)
         (* (* (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) t_m) (sin k)))
        (* t_m t_m)))
      (/ (/ (* (/ l k) l) (* (* k t_m) t_m)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.7e-58) {
		tmp = (2.0 / (fma((k / t_m), (k / t_m), 2.0) * tan(k))) * (((l / (k * t_m)) / t_m) * (l / t_m));
	} else if (k <= 1.7e+127) {
		tmp = l * (((l + l) / (((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) * t_m) * sin(k))) / (t_m * t_m));
	} else {
		tmp = (((l / k) * l) / ((k * t_m) * t_m)) / t_m;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.7e-58)
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * tan(k))) * Float64(Float64(Float64(l / Float64(k * t_m)) / t_m) * Float64(l / t_m)));
	elseif (k <= 1.7e+127)
		tmp = Float64(l * Float64(Float64(Float64(l + l) / Float64(Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) * t_m) * sin(k))) / Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * t_m) * t_m)) / t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 3.7e-58], N[(N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e+127], N[(l * N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.7000000000000003e-58

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 3.7000000000000003e-58 < k < 1.69999999999999989e127

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 57.5%

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

   if 1.69999999999999989e127 < k

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

Alternative 5: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \tan k} \cdot \left(\frac{\frac{\ell}{k \cdot t\_m}}{t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\ell}{\left(\sin k \cdot t\_m\right) \cdot t\_m}}{\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-143)
    (*
     (/ 2.0 (* (fma (/ k t_m) (/ k t_m) 2.0) (tan k)))
     (* (/ (/ l (* k t_m)) t_m) (/ l t_m)))
    (*
     l
     (/
      (* 2.0 (/ l (* (* (sin k) t_m) t_m)))
      (* (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-143) {
		tmp = (2.0 / (fma((k / t_m), (k / t_m), 2.0) * tan(k))) * (((l / (k * t_m)) / t_m) * (l / t_m));
	} else {
		tmp = l * ((2.0 * (l / ((sin(k) * t_m) * t_m))) / ((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-143)
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * tan(k))) * Float64(Float64(Float64(l / Float64(k * t_m)) / t_m) * Float64(l / t_m)));
	else
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l / Float64(Float64(sin(k) * t_m) * t_m))) / Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-143], N[(N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 * N[(l / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7999999999999999e-143

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 1.7999999999999999e-143 < t

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 63.5

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

   7. Applied rewrites 63.5%

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

Alternative 6: 71.0% accurate, 1.3× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.1e+96)
    (*
     (/ 2.0 (* (fma (/ k t_m) (/ k t_m) 2.0) (tan k)))
     (* (/ (/ l (* k t_m)) t_m) (/ l t_m)))
    (/ 2.0 (* (* (* t_m (* (* (/ t_m l) t_m) (/ (sin k) l))) (tan k)) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.1e+96) {
		tmp = (2.0 / (fma((k / t_m), (k / t_m), 2.0) * tan(k))) * (((l / (k * t_m)) / t_m) * (l / t_m));
	} else {
		tmp = 2.0 / (((t_m * (((t_m / l) * t_m) * (sin(k) / l))) * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.1e+96)
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * tan(k))) * Float64(Float64(Float64(l / Float64(k * t_m)) / t_m) * Float64(l / t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m / l) * t_m) * Float64(sin(k) / l))) * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 1.1e+96], N[(N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.0999999999999999e96

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 1.0999999999999999e96 < l

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 65.6%

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

Alternative 7: 70.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{k}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\frac{\ell}{\left(k \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4.6e+70)
    (/
     2.0
     (*
      (* (* t_m (* (* (/ t_m l) t_m) (/ k l))) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0)))
    (* (/ 1.0 k) (* (/ l (* (* k t_m) t_m)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.6e+70) {
		tmp = 2.0 / (((t_m * (((t_m / l) * t_m) * (k / l))) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4.6e+70)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m / l) * t_m) * Float64(k / l))) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(l / Float64(Float64(k * t_m) * t_m)) * Float64(l / t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 4.6e+70], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.59999999999999987e70

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. lift-fma.f64 73.1

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 66.7

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

   10. Applied rewrites 66.7%

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

   if 4.59999999999999987e70 < l

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 8: 68.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-262}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{t\_m \cdot t\_m}}{t\_m}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\_m\right)\right)\right) \cdot t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\frac{\ell}{\left(k \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.95e-262)
    (/ (* (/ l (sin k)) (/ (/ l k) (* t_m t_m))) t_m)
    (if (<= l 5e+37)
      (/
       (/
        (* (/ l k) l)
        (* (* k (+ t_m (* -0.16666666666666666 (* (pow k 2.0) t_m)))) t_m))
       t_m)
      (* (/ 1.0 k) (* (/ l (* (* k t_m) t_m)) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.95e-262) {
		tmp = ((l / sin(k)) * ((l / k) / (t_m * t_m))) / t_m;
	} else if (l <= 5e+37) {
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (pow(k, 2.0) * t_m)))) * t_m)) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.95d-262) then
        tmp = ((l / sin(k)) * ((l / k) / (t_m * t_m))) / t_m
    else if (l <= 5d+37) then
        tmp = (((l / k) * l) / ((k * (t_m + ((-0.16666666666666666d0) * ((k ** 2.0d0) * t_m)))) * t_m)) / t_m
    else
        tmp = (1.0d0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.95e-262) {
		tmp = ((l / Math.sin(k)) * ((l / k) / (t_m * t_m))) / t_m;
	} else if (l <= 5e+37) {
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (Math.pow(k, 2.0) * t_m)))) * t_m)) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.95e-262:
		tmp = ((l / math.sin(k)) * ((l / k) / (t_m * t_m))) / t_m
	elif l <= 5e+37:
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (math.pow(k, 2.0) * t_m)))) * t_m)) / t_m
	else:
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.95e-262)
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(Float64(l / k) / Float64(t_m * t_m))) / t_m);
	elseif (l <= 5e+37)
		tmp = Float64(Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m + Float64(-0.16666666666666666 * Float64((k ^ 2.0) * t_m)))) * t_m)) / t_m);
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(l / Float64(Float64(k * t_m) * t_m)) * Float64(l / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.95e-262)
		tmp = ((l / sin(k)) * ((l / k) / (t_m * t_m))) / t_m;
	elseif (l <= 5e+37)
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * ((k ^ 2.0) * t_m)))) * t_m)) / t_m;
	else
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 1.95e-262], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[l, 5e+37], N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m + N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.94999999999999992e-262

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}}{t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k}} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}{t} \]

      11. lower-/.f64 63.5

          \[\leadsto \frac{\frac{\ell}{\sin k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t}}}{t} \]

   10. Applied rewrites 63.5%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{t \cdot t}}}{t} \]

   if 1.94999999999999992e-262 < l < 4.99999999999999989e37

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 68.2

          \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot t}}{t} \]

   11. Applied rewrites 68.2%

       \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)\right)} \cdot t}}{t} \]

   if 4.99999999999999989e37 < l

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 9: 68.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.95 \cdot 10^{-262}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{\sin k \cdot t\_m}}{t\_m \cdot t\_m}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\_m\right)\right)\right) \cdot t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\frac{\ell}{\left(k \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.95e-262)
    (/ (* (/ l k) (/ l (* (sin k) t_m))) (* t_m t_m))
    (if (<= l 5e+37)
      (/
       (/
        (* (/ l k) l)
        (* (* k (+ t_m (* -0.16666666666666666 (* (pow k 2.0) t_m)))) t_m))
       t_m)
      (* (/ 1.0 k) (* (/ l (* (* k t_m) t_m)) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.95e-262) {
		tmp = ((l / k) * (l / (sin(k) * t_m))) / (t_m * t_m);
	} else if (l <= 5e+37) {
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (pow(k, 2.0) * t_m)))) * t_m)) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.95d-262) then
        tmp = ((l / k) * (l / (sin(k) * t_m))) / (t_m * t_m)
    else if (l <= 5d+37) then
        tmp = (((l / k) * l) / ((k * (t_m + ((-0.16666666666666666d0) * ((k ** 2.0d0) * t_m)))) * t_m)) / t_m
    else
        tmp = (1.0d0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.95e-262) {
		tmp = ((l / k) * (l / (Math.sin(k) * t_m))) / (t_m * t_m);
	} else if (l <= 5e+37) {
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (Math.pow(k, 2.0) * t_m)))) * t_m)) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.95e-262:
		tmp = ((l / k) * (l / (math.sin(k) * t_m))) / (t_m * t_m)
	elif l <= 5e+37:
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (math.pow(k, 2.0) * t_m)))) * t_m)) / t_m
	else:
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.95e-262)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / Float64(sin(k) * t_m))) / Float64(t_m * t_m));
	elseif (l <= 5e+37)
		tmp = Float64(Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m + Float64(-0.16666666666666666 * Float64((k ^ 2.0) * t_m)))) * t_m)) / t_m);
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(l / Float64(Float64(k * t_m) * t_m)) * Float64(l / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.95e-262)
		tmp = ((l / k) * (l / (sin(k) * t_m))) / (t_m * t_m);
	elseif (l <= 5e+37)
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * ((k ^ 2.0) * t_m)))) * t_m)) / t_m;
	else
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 1.95e-262], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+37], N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m + N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.94999999999999992e-262

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \ell}{\sin k \cdot t}}{t}}}{t} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \ell}{\sin k \cdot t}}{t \cdot t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\sin k \cdot t}}{\color{blue}{t \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \ell}{\sin k \cdot t}}{t \cdot t}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{\sin k \cdot t}}{t \cdot t}} \]

   if 1.94999999999999992e-262 < l < 4.99999999999999989e37

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 68.2

          \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot t}}{t} \]

   11. Applied rewrites 68.2%

       \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)\right)} \cdot t}}{t} \]

   if 4.99999999999999989e37 < l

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 10: 68.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\_m\right)\right)\right) \cdot t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\frac{\ell}{\left(k \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 5e+37)
    (/
     (/
      (* (/ l k) l)
      (* (* k (+ t_m (* -0.16666666666666666 (* (pow k 2.0) t_m)))) t_m))
     t_m)
    (* (/ 1.0 k) (* (/ l (* (* k t_m) t_m)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5e+37) {
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (pow(k, 2.0) * t_m)))) * t_m)) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 5d+37) then
        tmp = (((l / k) * l) / ((k * (t_m + ((-0.16666666666666666d0) * ((k ** 2.0d0) * t_m)))) * t_m)) / t_m
    else
        tmp = (1.0d0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5e+37) {
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (Math.pow(k, 2.0) * t_m)))) * t_m)) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 5e+37:
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * (math.pow(k, 2.0) * t_m)))) * t_m)) / t_m
	else:
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 5e+37)
		tmp = Float64(Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m + Float64(-0.16666666666666666 * Float64((k ^ 2.0) * t_m)))) * t_m)) / t_m);
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(l / Float64(Float64(k * t_m) * t_m)) * Float64(l / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 5e+37)
		tmp = (((l / k) * l) / ((k * (t_m + (-0.16666666666666666 * ((k ^ 2.0) * t_m)))) * t_m)) / t_m;
	else
		tmp = (1.0 / k) * ((l / ((k * t_m) * t_m)) * (l / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 5e+37], N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m + N[(-0.16666666666666666 * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(l / N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.99999999999999989e37

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 68.2

          \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)\right) \cdot t}}{t} \]

   11. Applied rewrites 68.2%

       \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{\left(k \cdot \left(t + -0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)\right)} \cdot t}}{t} \]

   if 4.99999999999999989e37 < l

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 11: 67.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot t\_m\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 19000000000000:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \ell}{t\_2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\frac{\ell}{t\_2} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (* k t_m) t_m)))
   (*
    t_s
    (if (<= l 19000000000000.0)
      (/ (/ (* (/ l k) l) t_2) t_m)
      (* (/ 1.0 k) (* (/ l t_2) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * t_m) * t_m;
	double tmp;
	if (l <= 19000000000000.0) {
		tmp = (((l / k) * l) / t_2) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / t_2) * (l / t_m));
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k * t_m) * t_m
    if (l <= 19000000000000.0d0) then
        tmp = (((l / k) * l) / t_2) / t_m
    else
        tmp = (1.0d0 / k) * ((l / t_2) * (l / t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * t_m) * t_m;
	double tmp;
	if (l <= 19000000000000.0) {
		tmp = (((l / k) * l) / t_2) / t_m;
	} else {
		tmp = (1.0 / k) * ((l / t_2) * (l / t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k * t_m) * t_m
	tmp = 0
	if l <= 19000000000000.0:
		tmp = (((l / k) * l) / t_2) / t_m
	else:
		tmp = (1.0 / k) * ((l / t_2) * (l / t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * t_m) * t_m)
	tmp = 0.0
	if (l <= 19000000000000.0)
		tmp = Float64(Float64(Float64(Float64(l / k) * l) / t_2) / t_m);
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(l / t_2) * Float64(l / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k * t_m) * t_m;
	tmp = 0.0;
	if (l <= 19000000000000.0)
		tmp = (((l / k) * l) / t_2) / t_m;
	else
		tmp = (1.0 / k) * ((l / t_2) * (l / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := Block[{t$95$2 = N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 19000000000000.0], N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(l / t$95$2), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9e13

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. 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-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.2

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

   3. Applied rewrites 66.2%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

   if 1.9e13 < l

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.5

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 12: 66.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((((l / k) * l) / ((k * t_m) * t_m)) / t_m)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((((l / k) * l) / ((k * t_m) * t_m)) / t_m);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((((l / k) * l) / ((k * t_m) * t_m)) / t_m)

Julia

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

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((((l / k) * l) / ((k * t_m) * t_m)) / t_m);
end

Wolfram

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_, k_] := N[(t$95$s * N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-*.f64 N/A

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

   2. 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-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow3 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 66.2

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

3. Applied rewrites 66.2%

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 63.4

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

8. Applied rewrites 63.4%

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

9. Taylor expanded in k around 0

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

    1. lower-*.f64 66.8

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

11. Applied rewrites 66.8%

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

Alternative 13: 63.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / ((t_m * (t_m * (k * t_m))) * k)) * l)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((t_m * (t_m * (k * t_m))) * k)) * l);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / ((t_m * (t_m * (k * t_m))) * k)) * l)

Julia

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

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / ((t_m * (t_m * (k * t_m))) * k)) * l);
end

Wolfram

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_, k_] := N[(t$95$s * N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 51.0

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

4. Applied rewrites 51.0%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow3 N/A

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

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   13. pow3 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   16. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   17. lower-*.f64 N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   18. lower-/.f64 55.5

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   19. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   20. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   21. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   22. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   23. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

6. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.6

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot \left(t \cdot k\right)\right) \cdot k} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot k} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

   9. lower-*.f64 63.2

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]

10. Applied rewrites 63.2%

    \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot k} \cdot \ell \]
11. Add Preprocessing

Alternative 14: 58.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* t_m (* (* t_m t_m) (* k k)))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * ((t_m * t_m) * (k * k)))) * l);
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (t_m * ((t_m * t_m) * (k * k)))) * l)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * ((t_m * t_m) * (k * k)))) * l);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (t_m * ((t_m * t_m) * (k * k)))) * l)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(t_m * Float64(Float64(t_m * t_m) * Float64(k * k)))) * l))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (t_m * ((t_m * t_m) * (k * k)))) * l);
end

Wolfram

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_, k_] := N[(t$95$s * N[(N[(l / N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   7. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   13. pow3 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   16. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   17. lower-*.f64 N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   18. lower-/.f64 55.5

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   19. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   20. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   21. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   22. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   23. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

6. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 59.6

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \color{blue}{\ell} \]

8. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   9. lower-*.f64 58.2

      \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

10. Applied rewrites 58.2%

    \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025157 
(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))))