Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 82.8%

Time: 8.0s

Alternatives: 18

Speedup: 12.5×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 4.8e-109)
    (/ 2.0 (/ (* (/ (pow (* (sin k) k) 2.0) l) (/ t_m l)) (cos k)))
    (if (<= t_m 1.4e+125)
      (/
       2.0
       (*
        (* (/ (* (* t_m t_m) (/ t_m l)) l) (sin k))
        (* (tan k) (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0))))
      (/
       2.0
       (/ (* (/ (* (pow (* (sin k) t_m) 2.0) 2.0) l) (/ t_m l)) (cos k)))))))

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 <= 4.8e-109) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / l) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.4e+125) {
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) / l) * sin(k)) * (tan(k) * ((pow((k / t_m), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / ((((pow((sin(k) * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 (t_m <= 4.8d-109) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / l) * (t_m / l)) / cos(k))
    else if (t_m <= 1.4d+125) then
        tmp = 2.0d0 / (((((t_m * t_m) * (t_m / l)) / l) * sin(k)) * (tan(k) * ((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0)))
    else
        tmp = 2.0d0 / ((((((sin(k) * t_m) ** 2.0d0) * 2.0d0) / l) * (t_m / l)) / cos(k))
    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 (t_m <= 4.8e-109) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / l) * (t_m / l)) / Math.cos(k));
	} else if (t_m <= 1.4e+125) {
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) / l) * Math.sin(k)) * (Math.tan(k) * ((Math.pow((k / t_m), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / ((((Math.pow((Math.sin(k) * t_m), 2.0) * 2.0) / l) * (t_m / l)) / Math.cos(k));
	}
	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 t_m <= 4.8e-109:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / l) * (t_m / l)) / math.cos(k))
	elif t_m <= 1.4e+125:
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) / l) * math.sin(k)) * (math.tan(k) * ((math.pow((k / t_m), 2.0) + 1.0) + 1.0)))
	else:
		tmp = 2.0 / ((((math.pow((math.sin(k) * t_m), 2.0) * 2.0) / l) * (t_m / l)) / math.cos(k))
	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 <= 4.8e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / l) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.4e+125)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l)) / l) * sin(k)) * Float64(tan(k) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(sin(k) * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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 (t_m <= 4.8e-109)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / l) * (t_m / l)) / cos(k));
	elseif (t_m <= 1.4e+125)
		tmp = 2.0 / (((((t_m * t_m) * (t_m / l)) / l) * sin(k)) * (tan(k) * ((((k / t_m) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = 2.0 / ((((((sin(k) * t_m) ^ 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	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[t$95$m, 4.8e-109], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+125], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.79999999999999977e-109

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 86.8%

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

   10. Taylor expanded in t around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. unpow-prod-down N/A

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

       8. lift-sin.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-pow.f64 68.8

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

   12. Applied rewrites 68.8%

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

   if 4.79999999999999977e-109 < t < 1.4e125

   1. Initial program 80.1%

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

   4. Applied rewrites 83.5%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 83.4

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

   6. Applied rewrites 83.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 88.1

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

   8. Applied rewrites 88.1%

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

   if 1.4e125 < t

   1. Initial program 70.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 92.0%

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

   10. Taylor expanded in t around inf

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. unpow-prod-down N/A

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

       10. lift-sin.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-pow.f64 92.0

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

   12. Applied rewrites 92.0%

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

Alternative 2: 84.5% accurate, 0.8× speedup

Math

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

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
  (/
   2.0
   (/
    (*
     (/ (fma (pow (* (sin k) t_m) 2.0) 2.0 (* (pow (sin k) 2.0) (* k k))) l)
     (/ t_m l))
    (cos k)))))

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 * (2.0 / (((fma(pow((sin(k) * t_m), 2.0), 2.0, (pow(sin(k), 2.0) * (k * k))) / l) * (t_m / l)) / cos(k)));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, Float64((sin(k) ^ 2.0) * Float64(k * k))) / l) * Float64(t_m / l)) / cos(k))))
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[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 77.2%

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

7. Applied rewrites 78.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 86.5%

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

    1. lift-pow.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-sin.f64 N/A

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

    4. unpow-prod-down N/A

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

    5. lower-*.f64 N/A

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

    6. lower-pow.f64 N/A

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

    7. lift-sin.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 86.5

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

11. Applied rewrites 86.5%

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

Alternative 3: 84.5% accurate, 0.8× speedup

Math

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

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
  (/
   2.0
   (/
    (*
     (/ (fma (pow (* (sin k) t_m) 2.0) 2.0 (pow (* (sin k) k) 2.0)) l)
     (/ t_m l))
    (cos k)))))

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 * (2.0 / (((fma(pow((sin(k) * t_m), 2.0), 2.0, pow((sin(k) * k), 2.0)) / l) * (t_m / l)) / cos(k)));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(fma((Float64(sin(k) * t_m) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / l) * Float64(t_m / l)) / cos(k))))
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[(2.0 / N[(N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 77.2%

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

7. Applied rewrites 78.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

9. Applied rewrites 86.5%

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

Alternative 4: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 7.5e-109)
    (/ 2.0 (/ (* (/ (pow (* (sin k) k) 2.0) l) (/ t_m l)) (cos k)))
    (if (<= t_m 1.05e+90)
      (/
       2.0
       (*
        (* (/ (/ (* (* t_m t_m) t_m) l) l) (sin k))
        (* (tan k) (fma (/ k t_m) (/ k t_m) 2.0))))
      (/ 2.0 (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k)))))))

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 <= 7.5e-109) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / l) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.05e+90) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l) / l) * sin(k)) * (tan(k) * fma((k / t_m), (k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 <= 7.5e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / l) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.05e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l) / l) * sin(k)) * Float64(tan(k) * fma(Float64(k / t_m), Float64(k / t_m), 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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, 7.5e-109], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+90], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.49999999999999982e-109

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 86.8%

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

   10. Taylor expanded in t around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. unpow-prod-down N/A

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

       8. lift-sin.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-pow.f64 68.8

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

   12. Applied rewrites 68.8%

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

   if 7.49999999999999982e-109 < t < 1.0499999999999999e90

   1. Initial program 84.3%

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

   4. Applied rewrites 88.1%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 88.1

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

   6. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 88.1

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

   8. Applied rewrites 88.1%

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

   if 1.0499999999999999e90 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 89.6

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

   12. Applied rewrites 89.6%

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

Alternative 5: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2, -0.6666666666666666, {\ell}^{-1}\right), k \cdot k, t\_2 \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \end{array} \]

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 (/ (* t_m t_m) l)))
   (*
    t_s
    (if (<= t_m 7.5e-109)
      (/
       2.0
       (/
        (*
         (*
          (fma (fma t_2 -0.6666666666666666 (pow l -1.0)) (* k k) (* t_2 2.0))
          (* k k))
         (/ t_m l))
        (cos k)))
      (if (<= t_m 1.05e+90)
        (/
         2.0
         (*
          (* (/ (/ (* (* t_m t_m) t_m) l) l) (sin k))
          (* (tan k) (fma (/ k t_m) (/ k t_m) 2.0))))
        (/
         2.0
         (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k))))))))

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 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 7.5e-109) {
		tmp = 2.0 / (((fma(fma(t_2, -0.6666666666666666, pow(l, -1.0)), (k * k), (t_2 * 2.0)) * (k * k)) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.05e+90) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l) / l) * sin(k)) * (tan(k) * fma((k / t_m), (k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 7.5e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(t_2, -0.6666666666666666, (l ^ -1.0)), Float64(k * k), Float64(t_2 * 2.0)) * Float64(k * k)) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.05e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l) / l) * sin(k)) * Float64(tan(k) * fma(Float64(k / t_m), Float64(k / t_m), 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-109], N[(2.0 / N[(N[(N[(N[(N[(t$95$2 * -0.6666666666666666 + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(t$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+90], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.49999999999999982e-109

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 86.8%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 50.3%

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

   if 7.49999999999999982e-109 < t < 1.0499999999999999e90

   1. Initial program 84.3%

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

   4. Applied rewrites 88.1%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 88.1

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

   6. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 88.1

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

   8. Applied rewrites 88.1%

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

   if 1.0499999999999999e90 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 89.6

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

   12. Applied rewrites 89.6%

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

Alternative 6: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 7.5e-109)
    (/
     2.0
     (/
      (*
       (/
        (*
         (fma
          (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
          k
          (* (* t_m t_m) 2.0))
         (* k k))
        l)
       (/ t_m l))
      (cos k)))
    (if (<= t_m 1.05e+90)
      (/
       2.0
       (*
        (* (/ (/ (* (* t_m t_m) t_m) l) l) (sin k))
        (* (tan k) (fma (/ k t_m) (/ k t_m) 2.0))))
      (/ 2.0 (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k)))))))

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 <= 7.5e-109) {
		tmp = 2.0 / ((((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / l) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.05e+90) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l) / l) * sin(k)) * (tan(k) * fma((k / t_m), (k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 <= 7.5e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / l) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.05e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l) / l) * sin(k)) * Float64(tan(k) * fma(Float64(k / t_m), Float64(k / t_m), 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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, 7.5e-109], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+90], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.49999999999999982e-109

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 46.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 54.4%

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

   if 7.49999999999999982e-109 < t < 1.0499999999999999e90

   1. Initial program 84.3%

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

   4. Applied rewrites 88.1%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 88.1

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

   6. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 88.1

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

   8. Applied rewrites 88.1%

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

   if 1.0499999999999999e90 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 89.6

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

   12. Applied rewrites 89.6%

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

Alternative 7: 78.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\frac{k \cdot k}{t\_m \cdot t\_m} + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 7.5e-109)
    (/
     2.0
     (/
      (*
       (/
        (*
         (fma
          (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
          k
          (* (* t_m t_m) 2.0))
         (* k k))
        l)
       (/ t_m l))
      (cos k)))
    (if (<= t_m 1.05e+90)
      (/
       2.0
       (*
        (* (/ (/ (* (* t_m t_m) t_m) l) l) (sin k))
        (* (tan k) (+ (/ (* k k) (* t_m t_m)) 2.0))))
      (/ 2.0 (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k)))))))

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 <= 7.5e-109) {
		tmp = 2.0 / ((((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / l) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.05e+90) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l) / l) * sin(k)) * (tan(k) * (((k * k) / (t_m * t_m)) + 2.0)));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 <= 7.5e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / l) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.05e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l) / l) * sin(k)) * Float64(tan(k) * Float64(Float64(Float64(k * k) / Float64(t_m * t_m)) + 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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, 7.5e-109], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+90], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.49999999999999982e-109

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 46.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 54.4%

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

   if 7.49999999999999982e-109 < t < 1.0499999999999999e90

   1. Initial program 84.3%

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

   4. Applied rewrites 88.1%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 88.1

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 86.3

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

   9. Applied rewrites 86.3%

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

   if 1.0499999999999999e90 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 89.6

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

   12. Applied rewrites 89.6%

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

Alternative 8: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 5.2e-83)
    (/
     2.0
     (/
      (*
       (/
        (*
         (fma
          (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
          k
          (* (* t_m t_m) 2.0))
         (* k k))
        l)
       (/ t_m l))
      (cos k)))
    (if (<= t_m 1.05e+90)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l l)) (sin k)) (tan k))
        (fma (/ k t_m) (/ k t_m) 2.0)))
      (/ 2.0 (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k)))))))

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 <= 5.2e-83) {
		tmp = 2.0 / ((((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / l) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.05e+90) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l * l)) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 <= 5.2e-83)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / l) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.05e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l * l)) * sin(k)) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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, 5.2e-83], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+90], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $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[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.20000000000000018e-83

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 47.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 55.5%

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

   if 5.20000000000000018e-83 < t < 1.0499999999999999e90

   1. Initial program 84.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 84.3

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

   6. Applied rewrites 84.3%

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

   if 1.0499999999999999e90 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 89.6

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

   12. Applied rewrites 89.6%

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

Alternative 9: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t\_m \cdot t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 5.2e-83)
    (/
     2.0
     (/
      (*
       (/
        (*
         (fma
          (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
          k
          (* (* t_m t_m) 2.0))
         (* k k))
        l)
       (/ t_m l))
      (cos k)))
    (if (<= t_m 1.05e+90)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (/ (* k k) (* t_m t_m))) 1.0)))
      (/ 2.0 (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k)))))))

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 <= 5.2e-83) {
		tmp = 2.0 / ((((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / l) * (t_m / l)) / cos(k));
	} else if (t_m <= 1.05e+90) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 <= 5.2e-83)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / l) * Float64(t_m / l)) / cos(k)));
	elseif (t_m <= 1.05e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k * k) / Float64(t_m * t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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, 5.2e-83], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+90], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.20000000000000018e-83

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 47.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 55.5%

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

   if 5.20000000000000018e-83 < t < 1.0499999999999999e90

   1. Initial program 84.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 84.3

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

   7. Applied rewrites 84.3%

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

   if 1.0499999999999999e90 < t

   1. Initial program 67.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 89.6

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

   12. Applied rewrites 89.6%

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

Alternative 10: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

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 4e-44)
    (/
     2.0
     (/
      (*
       (/
        (*
         (fma
          (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
          k
          (* (* t_m t_m) 2.0))
         (* k k))
        l)
       (/ t_m l))
      (cos k)))
    (/ 2.0 (/ (* (/ (* (pow (* k t_m) 2.0) 2.0) l) (/ t_m l)) (cos k))))))

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 <= 4e-44) {
		tmp = 2.0 / ((((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / l) * (t_m / l)) / cos(k));
	} else {
		tmp = 2.0 / ((((pow((k * t_m), 2.0) * 2.0) / l) * (t_m / l)) / cos(k));
	}
	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 <= 4e-44)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / l) * Float64(t_m / l)) / cos(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / l) * Float64(t_m / l)) / cos(k)));
	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, 4e-44], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999981e-44

   1. Initial program 55.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 49.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 56.8%

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

   if 3.99999999999999981e-44 < t

   1. Initial program 74.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 87.2%

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

   10. Taylor expanded in k around 0

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

       1. unpow-prod-down N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow-prod-down N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-*.f64 80.6

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

   12. Applied rewrites 80.6%

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

Alternative 11: 70.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3:\\ \;\;\;\;\frac{2}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell} \cdot \frac{t\_m}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

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 3.3)
    (/
     2.0
     (/
      (*
       (/
        (*
         (fma
          (* (fma (* t_m t_m) -0.6666666666666666 1.0) k)
          k
          (* (* t_m t_m) 2.0))
         (* k k))
        l)
       (/ t_m l))
      (cos k)))
    (/ (* l l) (* (pow (* k t_m) 2.0) 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 <= 3.3) {
		tmp = 2.0 / ((((fma((fma((t_m * t_m), -0.6666666666666666, 1.0) * k), k, ((t_m * t_m) * 2.0)) * (k * k)) / l) * (t_m / l)) / cos(k));
	} else {
		tmp = (l * l) / (pow((k * t_m), 2.0) * 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 <= 3.3)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) * Float64(k * k)) / l) * Float64(t_m / l)) / cos(k)));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k * t_m) ^ 2.0) * 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, 3.3], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.2999999999999998

   1. Initial program 55.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   10. Applied rewrites 49.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

   12. Applied rewrites 56.9%

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

   if 3.2999999999999998 < t

   1. Initial program 75.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 62.7

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

   5. Applied rewrites 62.7%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.6

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

   7. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 78.8

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

   9. Applied rewrites 78.8%

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

Alternative 12: 63.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.72 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.72e+50)
    (/ 2.0 (* (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) k) 2.0))
    (/ 2.0 (* (/ (* (* k k) t_m) (* l l)) (* k k))))))

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 <= 1.72e+50) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * k) * 2.0);
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	}
	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 (k <= 1.72d+50) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * k) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k * k) * t_m) / (l * l)) * (k * k))
    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 (k <= 1.72e+50) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * Math.sin(k)) * k) * 2.0);
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	}
	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 k <= 1.72e+50:
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * math.sin(k)) * k) * 2.0)
	else:
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k))
	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 <= 1.72e+50)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * k) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(l * l)) * Float64(k * k)));
	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 (k <= 1.72e+50)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * k) * 2.0);
	else
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	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[k, 1.72e+50], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.72e50

   1. Initial program 61.7%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 59.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 69.3

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

   10. Applied rewrites 69.3%

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

   if 1.72e50 < k

   1. Initial program 60.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.72 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.49:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t\_m}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

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 0.49)
    (/ 2.0 (* (* (* k k) (/ t_m (* l l))) (* k k)))
    (/ (* l l) (* (pow (* k t_m) 2.0) 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 <= 0.49) {
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k));
	} else {
		tmp = (l * l) / (pow((k * t_m), 2.0) * 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 (t_m <= 0.49d0) then
        tmp = 2.0d0 / (((k * k) * (t_m / (l * l))) * (k * k))
    else
        tmp = (l * l) / (((k * t_m) ** 2.0d0) * 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 (t_m <= 0.49) {
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k));
	} else {
		tmp = (l * l) / (Math.pow((k * t_m), 2.0) * 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 t_m <= 0.49:
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k))
	else:
		tmp = (l * l) / (math.pow((k * t_m), 2.0) * 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 <= 0.49)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m / Float64(l * l))) * Float64(k * k)));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k * t_m) ^ 2.0) * 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 (t_m <= 0.49)
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k));
	else
		tmp = (l * l) / (((k * t_m) ^ 2.0) * 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[t$95$m, 0.49], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 0.48999999999999999

   1. Initial program 55.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 60.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 58.7

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

   8. Applied rewrites 58.7%

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

   if 0.48999999999999999 < t

   1. Initial program 75.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 62.7

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

   5. Applied rewrites 62.7%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.6

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

   7. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 78.8

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

   9. Applied rewrites 78.8%

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

Alternative 14: 62.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-97}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.3333333333333333, 2\right), t\_m \cdot t\_m, k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.1e-97)
    (* l (/ l (* k (* k (pow t_m 3.0)))))
    (if (<= k 1.9e+71)
      (/
       2.0
       (*
        (/
         (* (fma (fma (* k k) 0.3333333333333333 2.0) (* t_m t_m) (* k k)) t_m)
         (* l l))
        (* k k)))
      (/ 2.0 (* (/ (* (* k k) t_m) (* l l)) (* k k)))))))

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 <= 1.1e-97) {
		tmp = l * (l / (k * (k * pow(t_m, 3.0))));
	} else if (k <= 1.9e+71) {
		tmp = 2.0 / (((fma(fma((k * k), 0.3333333333333333, 2.0), (t_m * t_m), (k * k)) * t_m) / (l * l)) * (k * k));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	}
	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 <= 1.1e-97)
		tmp = Float64(l * Float64(l / Float64(k * Float64(k * (t_m ^ 3.0)))));
	elseif (k <= 1.9e+71)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(k * k), 0.3333333333333333, 2.0), Float64(t_m * t_m), Float64(k * k)) * t_m) / Float64(l * l)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(l * l)) * Float64(k * k)));
	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, 1.1e-97], N[(l * N[(l / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.9e+71], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.0999999999999999e-97

   1. Initial program 61.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 53.0

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

   5. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 58.2

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

   7. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 65.0

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

   9. Applied rewrites 65.0%

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

   if 1.0999999999999999e-97 < k < 1.9e71

   1. Initial program 67.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 79.7

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

   8. Applied rewrites 79.7%

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

   if 1.9e71 < k

   1. Initial program 57.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

Alternative 15: 58.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-97}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.3333333333333333, 2\right), t\_m \cdot t\_m, k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.1e-97)
    (* l (/ l (* (* k k) (* (* t_m t_m) t_m))))
    (if (<= k 9e+72)
      (/
       2.0
       (*
        (/
         (* (fma (fma (* k k) 0.3333333333333333 2.0) (* t_m t_m) (* k k)) t_m)
         (* l l))
        (* k k)))
      (/ 2.0 (* (/ (* (* k k) t_m) (* l l)) (* k k)))))))

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 <= 1.1e-97) {
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	} else if (k <= 9e+72) {
		tmp = 2.0 / (((fma(fma((k * k), 0.3333333333333333, 2.0), (t_m * t_m), (k * k)) * t_m) / (l * l)) * (k * k));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	}
	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 <= 1.1e-97)
		tmp = Float64(l * Float64(l / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))));
	elseif (k <= 9e+72)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(k * k), 0.3333333333333333, 2.0), Float64(t_m * t_m), Float64(k * k)) * t_m) / Float64(l * l)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(l * l)) * Float64(k * k)));
	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, 1.1e-97], N[(l * N[(l / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9e+72], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.0999999999999999e-97

   1. Initial program 61.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 53.0

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

   5. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 58.2

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 58.2%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 58.2

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. Applied rewrites 58.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 1.0999999999999999e-97 < k < 8.9999999999999997e72

   1. Initial program 67.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 79.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{t \cdot \left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot \left(2 + \frac{1}{3} \cdot {k}^{2}\right) + {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 + \frac{1}{3} \cdot {k}^{2}\right) \cdot {t}^{2} + {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2 + \frac{1}{3} \cdot {k}^{2}, {t}^{2}, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{1}{3} \cdot {k}^{2} + 2, {t}^{2}, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({k}^{2} \cdot \frac{1}{3} + 2, {t}^{2}, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left({k}^{2}, \frac{1}{3}, 2\right), {t}^{2}, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, \frac{1}{3}, 2\right), {t}^{2}, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, \frac{1}{3}, 2\right), {t}^{2}, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, \frac{1}{3}, 2\right), t \cdot t, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, \frac{1}{3}, 2\right), t \cdot t, {k}^{2}\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, \frac{1}{3}, 2\right), t \cdot t, k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      13. lift-*.f64 79.7

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.3333333333333333, 2\right), t \cdot t, k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 79.7%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.3333333333333333, 2\right), t \cdot t, k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   if 8.9999999999999997e72 < k

   1. Initial program 57.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 63.1

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 5.8e+14)
    (* l (/ l (* (* k k) (* (* t_m t_m) t_m))))
    (/ 2.0 (* (/ (* (* k k) t_m) (* l l)) (* k k))))))

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 <= 5.8e+14) {
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	}
	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 (k <= 5.8d+14) then
        tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)))
    else
        tmp = 2.0d0 / ((((k * k) * t_m) / (l * l)) * (k * k))
    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 (k <= 5.8e+14) {
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	}
	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 k <= 5.8e+14:
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)))
	else:
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k))
	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 <= 5.8e+14)
		tmp = Float64(l * Float64(l / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(l * l)) * Float64(k * k)));
	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 (k <= 5.8e+14)
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	else
		tmp = 2.0 / ((((k * k) * t_m) / (l * l)) * (k * k));
	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[k, 5.8e+14], N[(l * N[(l / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e14

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 54.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 59.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 59.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 59.3

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. Applied rewrites 59.3%

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 5.8e14 < k

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 65.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 64.0

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 64.0%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t\_m}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 6e+14)
    (* l (/ l (* (* k k) (* (* t_m t_m) t_m))))
    (/ 2.0 (* (* (* k k) (/ t_m (* l l))) (* k k))))))

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 <= 6e+14) {
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k));
	}
	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 (k <= 6d+14) then
        tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)))
    else
        tmp = 2.0d0 / (((k * k) * (t_m / (l * l))) * (k * k))
    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 (k <= 6e+14) {
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	} else {
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k));
	}
	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 k <= 6e+14:
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)))
	else:
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k))
	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 <= 6e+14)
		tmp = Float64(l * Float64(l / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(t_m / Float64(l * l))) * Float64(k * k)));
	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 (k <= 6e+14)
		tmp = l * (l / ((k * k) * ((t_m * t_m) * t_m)));
	else
		tmp = 2.0 / (((k * k) * (t_m / (l * l))) * (k * k));
	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[k, 6e+14], N[(l * N[(l / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e14

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 54.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 59.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 59.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 59.3

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. Applied rewrites 59.3%

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 6e14 < k

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot \color{blue}{{k}^{2}}} \]

   5. Applied rewrites 65.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right), k \cdot k, 2 \cdot {t}^{3}\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

      7. lift-*.f64 63.9

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 63.9%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 55.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\right) \end{array} \]

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 (/ l (* (* k 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 * (l / ((k * 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 * (l / ((k * 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 * (l / ((k * 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 * (l / ((k * 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(l * Float64(l / Float64(Float64(k * k) * Float64(Float64(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 * (l / ((k * 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[(l * N[(l / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 55.1

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 55.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   3. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   7. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   13. lift-*.f64 59.2

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

7. Applied rewrites 59.2%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 59.2

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

9. Applied rewrites 59.2%

   \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025027 
(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))))