Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 79.5%

Time: 7.7s

Alternatives: 10

Speedup: 6.6×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% 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: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot \left(-\cos k\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(\sin k \cdot \left(-\sin k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k}{l\_m} \cdot \tan k\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<= t_m 3.2e-37)
      (*
       2.0
       (/
        (* (* l_m l_m) (- (cos k)))
        (* (* k k) (* t_m (* (sin k) (- (sin k)))))))
      (if (<= t_m 2.8e+129)
        (/
         2.0
         (* (* (/ (* (* (* t_m t_m) (/ t_m l_m)) (sin k)) l_m) (tan k)) t_2))
        (/
         2.0
         (*
          (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k)) (tan k))
          t_2)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if (t_m <= 3.2e-37) {
		tmp = 2.0 * (((l_m * l_m) * -cos(k)) / ((k * k) * (t_m * (sin(k) * -sin(k)))));
	} else if (t_m <= 2.8e+129) {
		tmp = 2.0 / ((((((t_m * t_m) * (t_m / l_m)) * sin(k)) / l_m) * tan(k)) * t_2);
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k)) * tan(k)) * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (t_m <= 3.2e-37)
		tmp = Float64(2.0 * Float64(Float64(Float64(l_m * l_m) * Float64(-cos(k))) / Float64(Float64(k * k) * Float64(t_m * Float64(sin(k) * Float64(-sin(k)))))));
	elseif (t_m <= 2.8e+129)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l_m)) * sin(k)) / l_m) * tan(k)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k)) * tan(k)) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e-37], N[(2.0 * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * (-N[Cos[k], $MachinePrecision])), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(N[Sin[k], $MachinePrecision] * (-N[Sin[k], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.8e+129], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.1999999999999999e-37

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-PI.f64 39.5

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

   6. Applied rewrites 39.5%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 60.6%

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

   if 3.1999999999999999e-37 < t < 2.79999999999999975e129

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.6

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

   3. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. exp-diff N/A

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

      9. pow-to-exp N/A

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

      10. pow-to-exp N/A

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

      11. pow2 N/A

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

      12. associate-/l/ N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. associate-*l/ N/A

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

   5. Applied rewrites 66.2%

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

   if 2.79999999999999975e129 < t

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.6

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

   3. Applied rewrites 70.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 70.6

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

   5. Applied rewrites 70.6%

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

Alternative 2: 69.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{l\_m}{k} \cdot \frac{l\_m}{\left(k \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 1150000000:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.3333333333333333, 2\right), t\_m \cdot t\_m, k \cdot k\right)}{l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k \cdot l\_m\right) \cdot l\_m\right)}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-73)
    (* (/ l_m k) (/ l_m (* (* k t_m) (* t_m t_m))))
    (if (<= k 1150000000.0)
      (/
       2.0
       (*
        t_m
        (/
         (/
          (*
           (* k k)
           (fma (fma (* k k) 0.3333333333333333 2.0) (* t_m t_m) (* k k)))
          l_m)
         l_m)))
      (/
       (* 2.0 (* (* (cos k) l_m) l_m))
       (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) (* k k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2e-73) {
		tmp = (l_m / k) * (l_m / ((k * t_m) * (t_m * t_m)));
	} else if (k <= 1150000000.0) {
		tmp = 2.0 / (t_m * ((((k * k) * fma(fma((k * k), 0.3333333333333333, 2.0), (t_m * t_m), (k * k))) / l_m) / l_m));
	} else {
		tmp = (2.0 * ((cos(k) * l_m) * l_m)) / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2e-73)
		tmp = Float64(Float64(l_m / k) * Float64(l_m / Float64(Float64(k * t_m) * Float64(t_m * t_m))));
	elseif (k <= 1150000000.0)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(Float64(k * k) * fma(fma(Float64(k * k), 0.3333333333333333, 2.0), Float64(t_m * t_m), Float64(k * k))) / l_m) / l_m)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) * l_m) * l_m)) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2e-73], N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / N[(N[(k * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1150000000.0], N[(2.0 / N[(t$95$m * N[(N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999999e-73

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if 1.99999999999999999e-73 < k < 1.15e9

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 56.2%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \left(t \cdot t\right) \cdot \left(2 - -0.3333333333333333 \cdot \left(k \cdot k\right)\right), \left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 61.4%

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

   if 1.15e9 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 155:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(l\_m \cdot l\_m\right) \cdot \cos k}{t\_m \cdot \mathsf{fma}\left(2, \left(t\_m \cdot t\_m\right) \cdot t\_2, \left(k \cdot k\right) \cdot t\_2\right)} \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (- 0.5 (* 0.5 (cos (* 2.0 k))))))
   (*
    t_s
    (if (<= k 155.0)
      (/
       2.0
       (*
        (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
        2.0))
      (*
       (/
        (* (* l_m l_m) (cos k))
        (* t_m (fma 2.0 (* (* t_m t_m) t_2) (* (* k k) t_2))))
       2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = 0.5 - (0.5 * cos((2.0 * k)));
	double tmp;
	if (k <= 155.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = (((l_m * l_m) * cos(k)) / (t_m * fma(2.0, ((t_m * t_m) * t_2), ((k * k) * t_2)))) * 2.0;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))
	tmp = 0.0
	if (k <= 155.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) * cos(k)) / Float64(t_m * fma(2.0, Float64(Float64(t_m * t_m) * t_2), Float64(Float64(k * k) * t_2)))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 155.0], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 155

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.6

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

   3. Applied rewrites 70.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 63.8%

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

   if 155 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\left(\frac{k \cdot k}{t \cdot t} + 2\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot 2} \]

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

   10. Applied rewrites 57.5%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{t \cdot \mathsf{fma}\left(2, \left(t \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right), \left(k \cdot k\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 80000:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k \cdot l\_m\right) \cdot l\_m\right)}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 80000.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      2.0))
    (/
     (* 2.0 (* (* (cos k) l_m) l_m))
     (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) (* k k))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 80000.0) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = (2.0 * ((cos(k) * l_m) * l_m)) / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 80000.0d0) then
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = (2.0d0 * ((cos(k) * l_m) * l_m)) / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * (k * k))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 80000.0) {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k) * l_m) * l_m)) / (((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * (k * k));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 80000.0:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = (2.0 * ((math.cos(k) * l_m) * l_m)) / (((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * (k * k))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 80000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) * l_m) * l_m)) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 80000.0)
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = (2.0 * ((cos(k) * l_m) * l_m)) / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 80000.0], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8e4

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 70.6

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

   3. Applied rewrites 70.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 63.8%

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

   if 8e4 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{l\_m}{k} \cdot \frac{l\_m}{\left(k \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\frac{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, 0.3333333333333333, 2\right), t\_m \cdot t\_m, k \cdot k\right)}{l\_m}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-73)
    (* (/ l_m k) (/ l_m (* (* k t_m) (* t_m t_m))))
    (if (<= k 6.5e+51)
      (/
       2.0
       (*
        t_m
        (/
         (/
          (*
           (* k k)
           (fma (fma (* k k) 0.3333333333333333 2.0) (* t_m t_m) (* k k)))
          l_m)
         l_m)))
      (/ 2.0 (* (/ (* (* k k) t_m) (* l_m l_m)) (* k k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2e-73) {
		tmp = (l_m / k) * (l_m / ((k * t_m) * (t_m * t_m)));
	} else if (k <= 6.5e+51) {
		tmp = 2.0 / (t_m * ((((k * k) * fma(fma((k * k), 0.3333333333333333, 2.0), (t_m * t_m), (k * k))) / l_m) / l_m));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / (l_m * l_m)) * (k * k));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2e-73)
		tmp = Float64(Float64(l_m / k) * Float64(l_m / Float64(Float64(k * t_m) * Float64(t_m * t_m))));
	elseif (k <= 6.5e+51)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(Float64(k * k) * fma(fma(Float64(k * k), 0.3333333333333333, 2.0), Float64(t_m * t_m), Float64(k * k))) / l_m) / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / Float64(l_m * l_m)) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2e-73], N[(N[(l$95$m / k), $MachinePrecision] * N[(l$95$m / N[(N[(k * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.5e+51], N[(2.0 / N[(t$95$m * N[(N[(N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.99999999999999999e-73

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if 1.99999999999999999e-73 < k < 6.5e51

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 56.2%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, \left(t \cdot t\right) \cdot \left(2 - -0.3333333333333333 \cdot \left(k \cdot k\right)\right), \left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{\ell \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 61.4%

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

   if 6.5e51 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]
   6. 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 54.1

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

   7. Applied rewrites 54.1%

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

Alternative 6: 64.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(t\_m \cdot t\_m\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-33}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_2, t\_m\right), k \cdot k, 2 \cdot t\_2\right)}{l\_m \cdot l\_m} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (* t_m t_m) t_m)))
   (*
    t_s
    (if (<= k 8.6e-33)
      (* l_m (/ l_m (* (* k (* t_m t_m)) (* k t_m))))
      (/
       2.0
       (*
        (/
         (fma (fma 0.3333333333333333 t_2 t_m) (* k k) (* 2.0 t_2))
         (* l_m l_m))
        (* k k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m * t_m) * t_m;
	double tmp;
	if (k <= 8.6e-33) {
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	} else {
		tmp = 2.0 / ((fma(fma(0.3333333333333333, t_2, t_m), (k * k), (2.0 * t_2)) / (l_m * l_m)) * (k * k));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m * t_m) * t_m)
	tmp = 0.0
	if (k <= 8.6e-33)
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * Float64(t_m * t_m)) * Float64(k * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(fma(0.3333333333333333, t_2, t_m), Float64(k * k), Float64(2.0 * t_2)) / Float64(l_m * l_m)) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 8.6e-33], N[(l$95$m * N[(l$95$m / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * t$95$2 + t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.60000000000000062e-33

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.6

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if 8.60000000000000062e-33 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

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

   4. Applied rewrites 57.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot t, t\right), k \cdot k, 2 \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 64.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}{l\_m \cdot l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.8e-9)
    (* l_m (/ l_m (* (* k (* t_m t_m)) (* k t_m))))
    (/ 2.0 (/ (* (* (* k k) (* k k)) t_m) (* l_m l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.8e-9) {
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / (l_m * l_m));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.8d-9) then
        tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)))
    else
        tmp = 2.0d0 / ((((k * k) * (k * k)) * t_m) / (l_m * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.8e-9) {
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / (l_m * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 3.8e-9:
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)))
	else:
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / (l_m * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.8e-9)
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * Float64(t_m * t_m)) * Float64(k * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * Float64(k * k)) * t_m) / Float64(l_m * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 3.8e-9)
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	else
		tmp = 2.0 / ((((k * k) * (k * k)) * t_m) / (l_m * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.8e-9], N[(l$95$m * N[(l$95$m / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.80000000000000011e-9

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.6

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if 3.80000000000000011e-9 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{\color{blue}{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      3. sqr-pow N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}{{\ell}^{2}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}\right) \cdot t}{{\ell}^{2}}} \]

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 51.8

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

   7. Applied rewrites 51.8%

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

Alternative 8: 64.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+22}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{l\_m \cdot l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e+22)
    (* l_m (/ l_m (* (* k (* t_m t_m)) (* k t_m))))
    (/ 2.0 (* t_m (/ (* (* (* k k) k) k) (* l_m l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.5e+22) {
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	} else {
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l_m * l_m)));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d+22) then
        tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)))
    else
        tmp = 2.0d0 / (t_m * ((((k * k) * k) * k) / (l_m * l_m)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.5e+22) {
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	} else {
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l_m * l_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.5e+22:
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)))
	else:
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l_m * l_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.5e+22)
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * Float64(t_m * t_m)) * Float64(k * t_m))));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(Float64(k * k) * k) * k) / Float64(l_m * l_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.5e+22)
		tmp = l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m)));
	else
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l_m * l_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.5e+22], N[(l$95$m * N[(l$95$m / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e22

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 54.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow3 N/A

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

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.8

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

   6. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.6

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if 1.5e22 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

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

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in t around 0

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

      1. sqr-pow N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}{\ell \cdot \ell}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}}{\ell \cdot \ell}} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 51.9

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

   10. Applied rewrites 51.9%

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

Alternative 9: 62.9% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m))))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m))))

Julia

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

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / ((k * (t_m * t_m)) * (k * t_m))));
end

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow3 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. pow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 54.8

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. pow3 N/A

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

   17. unpow3 N/A

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

   18. pow2 N/A

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

   19. associate-*l* N/A

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

   20. lower-*.f64 N/A

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

   21. pow2 N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f64 57.8

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

6. Applied rewrites 57.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 62.6

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 62.9

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

8. Applied rewrites 62.9%

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

Alternative 10: 55.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))))

Julia

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

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / 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 54.8%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. pow3 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. pow3 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 55.2

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

6. Applied rewrites 55.2%

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

Reproduce

herbie shell --seed 2025130 
(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))))