Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 79.4%

Time: 7.2s

Alternatives: 13

Speedup: 6.6×

• 28.6% 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: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.1e-33)
    (/ 2.0 (* k (* k (* (/ (pow (sin k) 2.0) (* (* (cos k) l_m) l_m)) t_m))))
    (/
     2.0
     (*
      (* (* (exp (fma 3.0 (log t_m) (* -2.0 (log l_m)))) (sin k)) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.1e-33) {
		tmp = 2.0 / (k * (k * ((pow(sin(k), 2.0) / ((cos(k) * l_m) * l_m)) * t_m)));
	} else {
		tmp = 2.0 / (((exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 6.1e-33)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64((sin(k) ^ 2.0) / Float64(Float64(cos(k) * l_m) * l_m)) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m)))) * sin(k)) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.1e-33], N[(2.0 / N[(k * N[(k * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.1000000000000001e-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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 56.1%

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

   8. Applied rewrites 58.5%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\frac{0.5 - \cos \left(k + k\right) \cdot 0.5}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot t\right)\right)}} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sin.f64 63.3

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

   10. Applied rewrites 63.3%

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

   if 6.1000000000000001e-33 < 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 70.7

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in t around 0

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

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

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 70.7

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

   8. Applied rewrites 70.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate--l+ N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 70.7

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

   10. Applied rewrites 70.7%

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

Alternative 2: 75.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.55 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.55e+22)
    (/ 2.0 (* k (* k (* (/ (pow (sin k) 2.0) (* (* (cos k) l_m) l_m)) t_m))))
    (/
     2.0
     (*
      (* (* (exp (fma 3.0 (log t_m) (* -2.0 (log l_m)))) (sin k)) (tan k))
      2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 3.55e+22) {
		tmp = 2.0 / (k * (k * ((pow(sin(k), 2.0) / ((cos(k) * l_m) * l_m)) * t_m)));
	} else {
		tmp = 2.0 / (((exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))) * sin(k)) * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 3.55e+22)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64((sin(k) ^ 2.0) / Float64(Float64(cos(k) * l_m) * l_m)) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m)))) * sin(k)) * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.55e+22], N[(2.0 / N[(k * N[(k * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5500000000000001e22

   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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 56.1%

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

   8. Applied rewrites 58.5%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\frac{0.5 - \cos \left(k + k\right) \cdot 0.5}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot t\right)\right)}} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sin.f64 63.3

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

   10. Applied rewrites 63.3%

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

   if 3.5500000000000001e22 < 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 70.7

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in t around 0

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

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

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 70.7

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

   6. Applied rewrites 70.7%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 64.2%

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

Alternative 3: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 58000000000000:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)} \cdot k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 58000000000000.0)
    (/
     2.0
     (*
      (* (* (exp (fma 3.0 (log t_m) (* -2.0 (log l_m)))) k) (tan k))
      (- (fma (/ k t_m) (/ k t_m) 1.0) -1.0)))
    (/
     2.0
     (*
      (/ (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) (* (* (cos k) l_m) l_m))
      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 <= 58000000000000.0) {
		tmp = 2.0 / (((exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))) * k) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) - -1.0));
	} else {
		tmp = 2.0 / (((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) / ((cos(k) * l_m) * l_m)) * 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 <= 58000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m)))) * k) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) - -1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) / Float64(Float64(cos(k) * l_m) * l_m)) * 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, 58000000000000.0], N[(2.0 / N[(N[(N[(N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e13

   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 70.7

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in t around 0

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

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

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 70.7

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

   8. Applied rewrites 70.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 64.8%

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

   if 5.8e13 < 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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 56.1%

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

   8. Applied rewrites 58.5%

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

   10. Applied rewrites 60.6%

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

Alternative 4: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 58000000000000:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)} \cdot k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(l\_m \cdot l\_m\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)}\\ \end{array} \end{array} \]

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 58000000000000.0)
    (/
     2.0
     (*
      (* (* (exp (fma 3.0 (log t_m) (* -2.0 (log l_m)))) k) (tan k))
      (- (fma (/ k t_m) (/ k t_m) 1.0) -1.0)))
    (*
     2.0
     (/
      (* (* l_m l_m) (cos k))
      (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ 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 <= 58000000000000.0) {
		tmp = 2.0 / (((exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))) * k) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) - -1.0));
	} else {
		tmp = 2.0 * (((l_m * l_m) * cos(k)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((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 <= 58000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m)))) * k) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) - -1.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l_m * l_m) * cos(k)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(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, 58000000000000.0], N[(2.0 / N[(N[(N[(N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e13

   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 70.7

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in t around 0

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

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

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 70.7

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

   8. Applied rewrites 70.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 64.8%

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

   if 5.8e13 < 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 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a-rev N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift--.f64 57.2

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 57.2

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

   7. Applied rewrites 57.2%

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

Alternative 5: 68.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 58000000000000:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)} \cdot k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k \cdot l\_m\right) \cdot l\_m\right)}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 58000000000000.0)
    (/
     2.0
     (*
      (* (* (exp (fma 3.0 (log t_m) (* -2.0 (log l_m)))) k) (tan k))
      (- (fma (/ k t_m) (/ k t_m) 1.0) -1.0)))
    (/
     (* 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 <= 58000000000000.0) {
		tmp = 2.0 / (((exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))) * k) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) - -1.0));
	} 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;
}

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 <= 58000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m)))) * k) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) - -1.0)));
	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

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, 58000000000000.0], N[(2.0 / N[(N[(N[(N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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 < 5.8e13

   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 70.7

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in t around 0

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

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

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 70.7

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

   8. Applied rewrites 70.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 64.8%

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

   if 5.8e13 < 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

   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 57.2%

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

Alternative 6: 68.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right)}\\ \mathbf{elif}\;t\_m \leq 10^{+100}:\\ \;\;\;\;\frac{l\_m}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)} \cdot k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) - -1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-23)
    (/ 2.0 (* (* k k) (* (* k k) (/ (/ t_m l_m) l_m))))
    (if (<= t_m 1e+100)
      (* (/ l_m (* k (* k (* (* t_m t_m) t_m)))) l_m)
      (/
       2.0
       (*
        (* (* (exp (fma 3.0 (log t_m) (* -2.0 (log l_m)))) k) (tan k))
        (- (fma (/ k t_m) (/ k t_m) 1.0) -1.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / ((k * k) * ((k * k) * ((t_m / l_m) / l_m)));
	} else if (t_m <= 1e+100) {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	} else {
		tmp = 2.0 / (((exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))) * k) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) - -1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.6e-23)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m))));
	elseif (t_m <= 1e+100)
		tmp = Float64(Float64(l_m / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))) * l_m);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m)))) * k) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) - -1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-23], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+100], N[(N[(l$95$m / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.60000000000000023e-23

   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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in k around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.7

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\ell \cdot \ell\right)}\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\ell} \cdot \ell\right)}\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 55.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 59.4

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

   12. Applied rewrites 59.4%

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

   if 7.60000000000000023e-23 < t < 1.00000000000000002e100

   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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 59.9

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

   10. Applied rewrites 59.9%

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

   if 1.00000000000000002e100 < 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 70.7

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

   3. Applied rewrites 70.7%

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

   4. Taylor expanded in t around 0

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

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

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 70.7

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

   6. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 70.7

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

   8. Applied rewrites 70.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 64.8%

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

Alternative 7: 67.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right) \cdot k}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right)}\\ \end{array} \end{array} \end{array} \]

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 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k)) t_2))
         2e+256)
      (/
       2.0
       (* (/ (* (* k (* (* t_m t_m) t_m)) k) (* (cos k) (* l_m l_m))) t_2))
      (/ 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 t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+256) {
		tmp = 2.0 / ((((k * ((t_m * t_m) * t_m)) * k) / (cos(k) * (l_m * l_m))) * t_2);
	} 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) :: t_2
    real(8) :: tmp
    t_2 = (1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2d+256) then
        tmp = 2.0d0 / ((((k * ((t_m * t_m) * t_m)) * k) / (cos(k) * (l_m * l_m))) * t_2)
    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 t_2 = (1.0 + Math.pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * t_2)) <= 2e+256) {
		tmp = 2.0 / ((((k * ((t_m * t_m) * t_m)) * k) / (Math.cos(k) * (l_m * l_m))) * t_2);
	} 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):
	t_2 = (1.0 + math.pow((k / t_m), 2.0)) + 1.0
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * t_2)) <= 2e+256:
		tmp = 2.0 / ((((k * ((t_m * t_m) * t_m)) * k) / (math.cos(k) * (l_m * l_m))) * t_2)
	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)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+256)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(Float64(t_m * t_m) * t_m)) * k) / Float64(cos(k) * Float64(l_m * l_m))) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * k) * Float64(Float64(t_m / 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)
	t_2 = (1.0 + ((k / t_m) ^ 2.0)) + 1.0;
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= 2e+256)
		tmp = 2.0 / ((((k * ((t_m * t_m) * t_m)) * k) / (cos(k) * (l_m * l_m))) * t_2);
	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_] := Block[{t$95$2 = N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2e+256], N[(2.0 / N[(N[(N[(N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e256

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-tan.f64 N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 54.9%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 52.8%

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

   if 2.0000000000000001e256 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in k around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.7

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\ell \cdot \ell\right)}\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\ell} \cdot \ell\right)}\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 55.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 59.4

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

   12. Applied rewrites 59.4%

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

Alternative 8: 66.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-23)
    (/ 2.0 (* (* k k) (* (* k k) (/ (/ t_m l_m) l_m))))
    (* (/ l_m (* k (* k (* (* t_m t_m) t_m)))) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / ((k * k) * ((k * k) * ((t_m / l_m) / l_m)));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d-23) then
        tmp = 2.0d0 / ((k * k) * ((k * k) * ((t_m / l_m) / l_m)))
    else
        tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / ((k * k) * ((k * k) * ((t_m / l_m) / l_m)));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.6e-23:
		tmp = 2.0 / ((k * k) * ((k * k) * ((t_m / l_m) / l_m)))
	else:
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.6e-23)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * k) * Float64(Float64(t_m / l_m) / l_m))));
	else
		tmp = Float64(Float64(l_m / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))) * l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.6e-23)
		tmp = 2.0 / ((k * k) * ((k * k) * ((t_m / l_m) / l_m)));
	else
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-23], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.60000000000000023e-23

   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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in k around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.7

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\ell \cdot \ell\right)}\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\ell} \cdot \ell\right)}\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 55.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 59.4

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

   12. Applied rewrites 59.4%

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

   if 7.60000000000000023e-23 < 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. 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 9: 64.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \frac{k \cdot k}{l\_m \cdot l\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-23)
    (/ 2.0 (* (* k k) (* t_m (/ (* k k) (* l_m l_m)))))
    (* (/ l_m (* k (* k (* (* t_m t_m) t_m)))) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / ((k * k) * (t_m * ((k * k) / (l_m * l_m))));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d-23) then
        tmp = 2.0d0 / ((k * k) * (t_m * ((k * k) / (l_m * l_m))))
    else
        tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / ((k * k) * (t_m * ((k * k) / (l_m * l_m))));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.6e-23:
		tmp = 2.0 / ((k * k) * (t_m * ((k * k) / (l_m * l_m))))
	else:
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.6e-23)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(t_m * Float64(Float64(k * k) / Float64(l_m * l_m)))));
	else
		tmp = Float64(Float64(l_m / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))) * l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.6e-23)
		tmp = 2.0 / ((k * k) * (t_m * ((k * k) / (l_m * l_m))));
	else
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-23], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(N[(k * k), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.60000000000000023e-23

   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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 54.6

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

   7. Applied rewrites 54.6%

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

   if 7.60000000000000023e-23 < 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. 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 10: 64.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_m}{l\_m \cdot l\_m} \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-23)
    (/ 2.0 (* k (* k (* (/ t_m (* l_m l_m)) (* k k)))))
    (* (/ l_m (* k (* k (* (* t_m t_m) t_m)))) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / (k * (k * ((t_m / (l_m * l_m)) * (k * k))));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d-23) then
        tmp = 2.0d0 / (k * (k * ((t_m / (l_m * l_m)) * (k * k))))
    else
        tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / (k * (k * ((t_m / (l_m * l_m)) * (k * k))));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.6e-23:
		tmp = 2.0 / (k * (k * ((t_m / (l_m * l_m)) * (k * k))))
	else:
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.6e-23)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_m / Float64(l_m * l_m)) * Float64(k * k)))));
	else
		tmp = Float64(Float64(l_m / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))) * l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.6e-23)
		tmp = 2.0 / (k * (k * ((t_m / (l_m * l_m)) * (k * k))));
	else
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-23], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.60000000000000023e-23

   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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in k around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.7

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\ell \cdot \ell\right)}\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\left(1 - 0.5 \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{\ell} \cdot \ell\right)}\right)} \]

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 55.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 55.6

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. pow2 N/A

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

       11. *-commutative N/A

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

   12. Applied rewrites 55.6%

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

   if 7.60000000000000023e-23 < 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. 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 11: 63.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-23)
    (/ 2.0 (* k (* (* (* k k) k) (/ t_m (* l_m l_m)))))
    (* (/ l_m (* k (* k (* (* t_m t_m) t_m)))) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / (k * (((k * k) * k) * (t_m / (l_m * l_m))));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d-23) then
        tmp = 2.0d0 / (k * (((k * k) * k) * (t_m / (l_m * l_m))))
    else
        tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.6e-23) {
		tmp = 2.0 / (k * (((k * k) * k) * (t_m / (l_m * l_m))));
	} else {
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.6e-23:
		tmp = 2.0 / (k * (((k * k) * k) * (t_m / (l_m * l_m))))
	else:
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.6e-23)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k * k) * k) * Float64(t_m / Float64(l_m * l_m)))));
	else
		tmp = Float64(Float64(l_m / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))) * l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.6e-23)
		tmp = 2.0 / (k * (((k * k) * k) * (t_m / (l_m * l_m))));
	else
		tmp = (l_m / (k * (k * ((t_m * t_m) * t_m)))) * l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-23], N[(2.0 / N[(k * N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.60000000000000023e-23

   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 t around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 56.1

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

   4. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 56.1%

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

   8. Applied rewrites 58.5%

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

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. unpow3 N/A

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

       4. pow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 53.9

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

   11. Applied rewrites 53.9%

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

   if 7.60000000000000023e-23 < 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. 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 59.9

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

   10. Applied rewrites 59.9%

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

Alternative 12: 61.6% accurate, 0.9× speedup

Math

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

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       INFINITY)
    (* (/ l_m (* k (* k (* (* t_m t_m) t_m)))) l_m)
    (* (/ l_m (* (* (* k k) (* t_m t_m)) t_m)) l_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 59.9

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

   10. Applied rewrites 59.9%

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

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 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 51.1

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

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
   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 55.3

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

   6. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow3 N/A

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

      7. unpow3 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 58.1

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

   10. Applied rewrites 58.1%

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

Alternative 13: 59.9% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(N[(l$95$m / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $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 51.1

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

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
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 55.3

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

6. Applied rewrites 55.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \color{blue}{\ell} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   13. lift-/.f64 55.3

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

8. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \cdot \ell \]

   5. pow3 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   9. pow3 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   11. lift-*.f64 59.9

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

10. Applied rewrites 59.9%

    \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))