Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.9% → 92.6%

Time: 7.5s

Alternatives: 22

Speedup: 6.6×

• 73.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 53.9% accurate, 1.0× speedup

Math

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

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: 92.6% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 1.55 \cdot 10^{+114}:\\ \;\;\;\;\left(\frac{2}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot \ell\right) \cdot \frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left|t\right|}{\ell} \cdot \left(\left(\left(\left|t\right| \cdot \frac{\sin k}{\ell}\right) \cdot \tan k\right) \cdot \left|t\right|\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{\left|t\right|}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 3.8e-29)
    (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) (* (tan k) (sin k))))
    (if (<= (fabs t) 1.55e+114)
      (*
       (* (/ 2.0 (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) l)
       (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))))
      (/
       2.0
       (*
        (* (/ (fabs t) l) (* (* (* (fabs t) (/ (sin k) l)) (tan k)) (fabs t)))
        (+ (+ 1.0 (pow (/ k (fabs t)) 2.0)) 1.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 3.8e-29) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * (tan(k) * sin(k)));
	} else if (fabs(t) <= 1.55e+114) {
		tmp = ((2.0 / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * l) * (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else {
		tmp = 2.0 / (((fabs(t) / l) * (((fabs(t) * (sin(k) / l)) * tan(k)) * fabs(t))) * ((1.0 + pow((k / fabs(t)), 2.0)) + 1.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 3.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * Float64(tan(k) * sin(k))));
	elseif (abs(t) <= 1.55e+114)
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * l) * Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(abs(t) / l) * Float64(Float64(Float64(abs(t) * Float64(sin(k) / l)) * tan(k)) * abs(t))) * Float64(Float64(1.0 + (Float64(k / abs(t)) ^ 2.0)) + 1.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-29], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.55e+114], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.79999999999999976e-29

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 3.79999999999999976e-29 < t < 1.55e114

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 55.5%

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

   if 1.55e114 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.3%

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 69.7%

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

   5. Applied rewrites 69.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 73.5%

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

   7. Applied rewrites 73.5%

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

Alternative 2: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \tan k \cdot \sin k\\ t_2 := \frac{k}{\left|t\right|}\\ t_3 := \frac{\left|t\right|}{\ell}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot t\_1}\\ \mathbf{elif}\;\left|t\right| \leq 1.55 \cdot 10^{+114}:\\ \;\;\;\;\left(\frac{2}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot \ell\right) \cdot \frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_2, t\_2, 2\right) \cdot \left(t\_3 \cdot \left(t\_3 \cdot \left|t\right|\right)\right)} \cdot \frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) (sin k))) (t_2 (/ k (fabs t))) (t_3 (/ (fabs t) l)))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.8e-29)
      (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) t_1))
      (if (<= (fabs t) 1.55e+114)
        (*
         (* (/ 2.0 (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) l)
         (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))))
        (*
         (/ 2.0 (* (fma t_2 t_2 2.0) (* t_3 (* t_3 (fabs t)))))
         (/ 1.0 t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * sin(k);
	double t_2 = k / fabs(t);
	double t_3 = fabs(t) / l;
	double tmp;
	if (fabs(t) <= 3.8e-29) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * t_1);
	} else if (fabs(t) <= 1.55e+114) {
		tmp = ((2.0 / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * l) * (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else {
		tmp = (2.0 / (fma(t_2, t_2, 2.0) * (t_3 * (t_3 * fabs(t))))) * (1.0 / t_1);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * sin(k))
	t_2 = Float64(k / abs(t))
	t_3 = Float64(abs(t) / l)
	tmp = 0.0
	if (abs(t) <= 3.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * t_1));
	elseif (abs(t) <= 1.55e+114)
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * l) * Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))));
	else
		tmp = Float64(Float64(2.0 / Float64(fma(t_2, t_2, 2.0) * Float64(t_3 * Float64(t_3 * abs(t))))) * Float64(1.0 / t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-29], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.55e+114], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$2 * t$95$2 + 2.0), $MachinePrecision] * N[(t$95$3 * N[(t$95$3 * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.79999999999999976e-29

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 3.79999999999999976e-29 < t < 1.55e114

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 55.5%

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

   if 1.55e114 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 67.0%

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

   5. Applied rewrites 67.0%

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

Alternative 3: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\\ t_2 := \tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2.9 \cdot 10^{+118}:\\ \;\;\;\;\left(\frac{2}{t\_2} \cdot \ell\right) \cdot \frac{\ell}{t\_1 \cdot \left|t\right|}\\ \mathbf{elif}\;\left|t\right| \leq 5.2 \cdot 10^{+197}:\\ \;\;\;\;\frac{\left(\frac{\frac{\ell}{\left|t\right|}}{t\_1} \cdot \ell\right) \cdot 2}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* (sin k) (fabs t)) (fabs t)))
        (t_2 (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.8e-29)
      (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) (* (tan k) (sin k))))
      (if (<= (fabs t) 2.9e+118)
        (* (* (/ 2.0 t_2) l) (/ l (* t_1 (fabs t))))
        (if (<= (fabs t) 5.2e+197)
          (/ (* (* (/ (/ l (fabs t)) t_1) l) 2.0) t_2)
          (/
           2.0
           (*
            (*
             (fabs t)
             (* (* (/ (fabs t) l) (fabs t)) (* (tan k) (/ (sin k) l))))
            2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = (sin(k) * fabs(t)) * fabs(t);
	double t_2 = tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0);
	double tmp;
	if (fabs(t) <= 3.8e-29) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * (tan(k) * sin(k)));
	} else if (fabs(t) <= 2.9e+118) {
		tmp = ((2.0 / t_2) * l) * (l / (t_1 * fabs(t)));
	} else if (fabs(t) <= 5.2e+197) {
		tmp = ((((l / fabs(t)) / t_1) * l) * 2.0) / t_2;
	} else {
		tmp = 2.0 / ((fabs(t) * (((fabs(t) / l) * fabs(t)) * (tan(k) * (sin(k) / l)))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(sin(k) * abs(t)) * abs(t))
	t_2 = Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))
	tmp = 0.0
	if (abs(t) <= 3.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * Float64(tan(k) * sin(k))));
	elseif (abs(t) <= 2.9e+118)
		tmp = Float64(Float64(Float64(2.0 / t_2) * l) * Float64(l / Float64(t_1 * abs(t))));
	elseif (abs(t) <= 5.2e+197)
		tmp = Float64(Float64(Float64(Float64(Float64(l / abs(t)) / t_1) * l) * 2.0) / t_2);
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(Float64(abs(t) / l) * abs(t)) * Float64(tan(k) * Float64(sin(k) / l)))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-29], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.9e+118], N[(N[(N[(2.0 / t$95$2), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 5.2e+197], N[(N[(N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.79999999999999976e-29

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 3.79999999999999976e-29 < t < 2.90000000000000016e118

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 55.5%

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

   if 2.90000000000000016e118 < t < 5.19999999999999975e197

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 59.6%

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

   7. Applied rewrites 59.6%

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

   if 5.19999999999999975e197 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.3%

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 69.7%

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 62.4%

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

Alternative 4: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{k}{\left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2.5 \cdot 10^{+155}:\\ \;\;\;\;\left(\frac{2}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot \ell\right) \cdot \frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(t\_1, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 3.8e-29)
      (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) (* (tan k) (sin k))))
      (if (<= (fabs t) 2.5e+155)
        (*
         (* (/ 2.0 (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) l)
         (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))))
        (/
         2.0
         (*
          (*
           (fabs t)
           (* (* (/ (fabs t) l) (fabs t)) (* (tan k) (/ (sin k) l))))
          (fma t_1 t_1 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = k / fabs(t);
	double tmp;
	if (fabs(t) <= 3.8e-29) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * (tan(k) * sin(k)));
	} else if (fabs(t) <= 2.5e+155) {
		tmp = ((2.0 / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * l) * (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else {
		tmp = 2.0 / ((fabs(t) * (((fabs(t) / l) * fabs(t)) * (tan(k) * (sin(k) / l)))) * fma(t_1, t_1, 2.0));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / abs(t))
	tmp = 0.0
	if (abs(t) <= 3.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * Float64(tan(k) * sin(k))));
	elseif (abs(t) <= 2.5e+155)
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * l) * Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(Float64(abs(t) / l) * abs(t)) * Float64(tan(k) * Float64(sin(k) / l)))) * fma(t_1, t_1, 2.0)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-29], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.5e+155], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.79999999999999976e-29

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 3.79999999999999976e-29 < t < 2.5e155

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 55.5%

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

   if 2.5e155 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.3%

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 69.7%

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

   5. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 69.7%

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

   7. Applied rewrites 69.7%

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

Alternative 5: 87.6% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{2}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)} \cdot \ell\right) \cdot \frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 3.8e-29)
    (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) (* (tan k) (sin k))))
    (if (<= (fabs t) 1.25e+150)
      (*
       (* (/ 2.0 (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0))) l)
       (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t))))
      (/
       2.0
       (*
        (* (fabs t) (* (* (/ (fabs t) l) (fabs t)) (* (tan k) (/ (sin k) l))))
        2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 3.8e-29) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * (tan(k) * sin(k)));
	} else if (fabs(t) <= 1.25e+150) {
		tmp = ((2.0 / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))) * l) * (l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else {
		tmp = 2.0 / ((fabs(t) * (((fabs(t) / l) * fabs(t)) * (tan(k) * (sin(k) / l)))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 3.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * Float64(tan(k) * sin(k))));
	elseif (abs(t) <= 1.25e+150)
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0))) * l) * Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(Float64(abs(t) / l) * abs(t)) * Float64(tan(k) * Float64(sin(k) / l)))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-29], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e+150], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.79999999999999976e-29

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 3.79999999999999976e-29 < t < 1.25000000000000002e150

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 55.5%

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

   if 1.25000000000000002e150 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.3%

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 69.7%

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 62.4%

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

Alternative 6: 87.3% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{\left(\left(\sin k \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|} \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(k, \frac{k}{\left|t\right| \cdot \left|t\right|}, 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)\right) \cdot 2}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 3.8e-29)
    (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) (* (tan k) (sin k))))
    (if (<= (fabs t) 1.25e+150)
      (*
       l
       (*
        (/ l (* (* (* (sin k) (fabs t)) (fabs t)) (fabs t)))
        (/ 2.0 (* (tan k) (fma k (/ k (* (fabs t) (fabs t))) 2.0)))))
      (/
       2.0
       (*
        (* (fabs t) (* (* (/ (fabs t) l) (fabs t)) (* (tan k) (/ (sin k) l))))
        2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 3.8e-29) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * (tan(k) * sin(k)));
	} else if (fabs(t) <= 1.25e+150) {
		tmp = l * ((l / (((sin(k) * fabs(t)) * fabs(t)) * fabs(t))) * (2.0 / (tan(k) * fma(k, (k / (fabs(t) * fabs(t))), 2.0))));
	} else {
		tmp = 2.0 / ((fabs(t) * (((fabs(t) / l) * fabs(t)) * (tan(k) * (sin(k) / l)))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 3.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * Float64(tan(k) * sin(k))));
	elseif (abs(t) <= 1.25e+150)
		tmp = Float64(l * Float64(Float64(l / Float64(Float64(Float64(sin(k) * abs(t)) * abs(t)) * abs(t))) * Float64(2.0 / Float64(tan(k) * fma(k, Float64(k / Float64(abs(t) * abs(t))), 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(Float64(abs(t) / l) * abs(t)) * Float64(tan(k) * Float64(sin(k) / l)))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 3.8e-29], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.25e+150], N[(l * N[(N[(l / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.79999999999999976e-29

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 3.79999999999999976e-29 < t < 1.25000000000000002e150

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 55.5%

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

   if 1.25000000000000002e150 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.3%

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 69.7%

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 62.4%

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

Alternative 7: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 9.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t\_1}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;\left|t\right| \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\left(\frac{\left|t\right|}{\ell} \cdot \left|t\right|\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 9.6e-26)
      (/ 2.0 (* (* (/ k l) (/ t_1 l)) (* (tan k) (sin k))))
      (if (<= (fabs t) 3.5e+152)
        (* (/ l (* (* (fabs t) (fabs t)) k)) (/ l t_1))
        (/
         2.0
         (*
          (*
           (fabs t)
           (* (* (/ (fabs t) l) (fabs t)) (* (tan k) (/ (sin k) l))))
          2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 9.6e-26) {
		tmp = 2.0 / (((k / l) * (t_1 / l)) * (tan(k) * sin(k)));
	} else if (fabs(t) <= 3.5e+152) {
		tmp = (l / ((fabs(t) * fabs(t)) * k)) * (l / t_1);
	} else {
		tmp = 2.0 / ((fabs(t) * (((fabs(t) / l) * fabs(t)) * (tan(k) * (sin(k) / l)))) * 2.0);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 9.6e-26) {
		tmp = 2.0 / (((k / l) * (t_1 / l)) * (Math.tan(k) * Math.sin(k)));
	} else if (Math.abs(t) <= 3.5e+152) {
		tmp = (l / ((Math.abs(t) * Math.abs(t)) * k)) * (l / t_1);
	} else {
		tmp = 2.0 / ((Math.abs(t) * (((Math.abs(t) / l) * Math.abs(t)) * (Math.tan(k) * (Math.sin(k) / l)))) * 2.0);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 9.6e-26:
		tmp = 2.0 / (((k / l) * (t_1 / l)) * (math.tan(k) * math.sin(k)))
	elif math.fabs(t) <= 3.5e+152:
		tmp = (l / ((math.fabs(t) * math.fabs(t)) * k)) * (l / t_1)
	else:
		tmp = 2.0 / ((math.fabs(t) * (((math.fabs(t) / l) * math.fabs(t)) * (math.tan(k) * (math.sin(k) / l)))) * 2.0)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 9.6e-26)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t_1 / l)) * Float64(tan(k) * sin(k))));
	elseif (abs(t) <= 3.5e+152)
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) * Float64(l / t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(Float64(Float64(abs(t) / l) * abs(t)) * Float64(tan(k) * Float64(sin(k) / l)))) * 2.0));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 9.6e-26)
		tmp = 2.0 / (((k / l) * (t_1 / l)) * (tan(k) * sin(k)));
	elseif (abs(t) <= 3.5e+152)
		tmp = (l / ((abs(t) * abs(t)) * k)) * (l / t_1);
	else
		tmp = 2.0 / ((abs(t) * (((abs(t) / l) * abs(t)) * (tan(k) * (sin(k) / l)))) * 2.0);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 9.6e-26], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 3.5e+152], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 9.6000000000000004e-26

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 9.6000000000000004e-26 < t < 3.49999999999999981e152

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 50.4%

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

   4. Applied rewrites 50.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 54.8%

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.0%

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.0%

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

   6. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.1%

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

   8. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 63.9%

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

   10. Applied rewrites 63.9%

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

   if 3.49999999999999981e152 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 60.3%

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 69.7%

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 62.4%

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

Alternative 8: 86.5% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot \left|t\right|}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 6.6e+35)
    (/ 2.0 (* (* (/ k l) (/ (* k (fabs t)) l)) (* (tan k) (sin k))))
    (* (/ 1.0 k) (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 6.6e+35) {
		tmp = 2.0 / (((k / l) * ((k * fabs(t)) / l)) * (tan(k) * sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 6.6e+35) {
		tmp = 2.0 / (((k / l) * ((k * Math.abs(t)) / l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 6.6e+35:
		tmp = 2.0 / (((k / l) * ((k * math.fabs(t)) / l)) * (math.tan(k) * math.sin(k)))
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 6.6e+35)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * abs(t)) / l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 6.6e+35)
		tmp = 2.0 / (((k / l) * ((k * abs(t)) / l)) * (tan(k) * sin(k)));
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6.6e+35], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.6000000000000003e35

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

   if 6.6000000000000003e35 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 54.4%

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

   4. Taylor expanded in k around 0

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

      1. lower-/.f64 53.2%

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

   6. Applied rewrites 53.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

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

Alternative 9: 85.3% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(\left|t\right| \cdot \frac{k}{\ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 6.6e+35)
    (/ 2.0 (* (* (/ k l) (* (fabs t) (/ k l))) (* (tan k) (sin k))))
    (* (/ 1.0 k) (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 6.6e+35) {
		tmp = 2.0 / (((k / l) * (fabs(t) * (k / l))) * (tan(k) * sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 6.6e+35) {
		tmp = 2.0 / (((k / l) * (Math.abs(t) * (k / l))) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 6.6e+35:
		tmp = 2.0 / (((k / l) * (math.fabs(t) * (k / l))) * (math.tan(k) * math.sin(k)))
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 6.6e+35)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(abs(t) * Float64(k / l))) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 6.6e+35)
		tmp = 2.0 / (((k / l) * (abs(t) * (k / l))) * (tan(k) * sin(k)));
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6.6e+35], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.6000000000000003e35

   1. Initial program 53.9%

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

   2. 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.4%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l/ N/A

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

   6. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.4%

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

   8. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 76.3%

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

   10. Applied rewrites 76.3%

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

   if 6.6000000000000003e35 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.9% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 68000000:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot \frac{\frac{\left|t\right|}{\ell}}{\ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 68000000.0)
    (/ 2.0 (* (* k (* k (/ (/ (fabs t) l) l))) (* (tan k) (sin k))))
    (* (/ 1.0 k) (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 68000000.0) {
		tmp = 2.0 / ((k * (k * ((fabs(t) / l) / l))) * (tan(k) * sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 68000000.0) {
		tmp = 2.0 / ((k * (k * ((Math.abs(t) / l) / l))) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 68000000.0:
		tmp = 2.0 / ((k * (k * ((math.fabs(t) / l) / l))) * (math.tan(k) * math.sin(k)))
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 68000000.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * Float64(Float64(abs(t) / l) / l))) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 68000000.0)
		tmp = 2.0 / ((k * (k * ((abs(t) / l) / l))) * (tan(k) * sin(k)));
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 68000000.0], N[(2.0 / N[(N[(k * N[(k * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.8e7

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \color{blue}{\ell}\right) \cdot \cos k}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(\frac{\sin k}{\cos k} \cdot \color{blue}{\sin k}\right)} \]

   6. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      8. lower-/.f64 64.5%

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   8. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      3. associate-/r* N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      5. lower-/.f64 69.2%

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   10. Applied rewrites 69.2%

       \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   if 6.8e7 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 3.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{\left|k\right|} \cdot \frac{\frac{\ell}{t\_1 \cdot t} \cdot \frac{\ell}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left|k\right| \cdot t\right) \cdot \left|k\right|}{\ell \cdot \ell} \cdot \left(\tan \left(\left|k\right|\right) \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))))
   (if (<= (fabs k) 3.7e-40)
     (* (/ 1.0 (fabs k)) (/ (* (/ l (* t_1 t)) (/ l t)) t))
     (/
      2.0
      (* (/ (* (* (fabs k) t) (fabs k)) (* l l)) (* (tan (fabs k)) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 3.7e-40) {
		tmp = (1.0 / fabs(k)) * (((l / (t_1 * t)) * (l / t)) / t);
	} else {
		tmp = 2.0 / ((((fabs(k) * t) * fabs(k)) / (l * l)) * (tan(fabs(k)) * t_1));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k))
    if (abs(k) <= 3.7d-40) then
        tmp = (1.0d0 / abs(k)) * (((l / (t_1 * t)) * (l / t)) / t)
    else
        tmp = 2.0d0 / ((((abs(k) * t) * abs(k)) / (l * l)) * (tan(abs(k)) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 3.7e-40) {
		tmp = (1.0 / Math.abs(k)) * (((l / (t_1 * t)) * (l / t)) / t);
	} else {
		tmp = 2.0 / ((((Math.abs(k) * t) * Math.abs(k)) / (l * l)) * (Math.tan(Math.abs(k)) * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 3.7e-40:
		tmp = (1.0 / math.fabs(k)) * (((l / (t_1 * t)) * (l / t)) / t)
	else:
		tmp = 2.0 / ((((math.fabs(k) * t) * math.fabs(k)) / (l * l)) * (math.tan(math.fabs(k)) * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 3.7e-40)
		tmp = Float64(Float64(1.0 / abs(k)) * Float64(Float64(Float64(l / Float64(t_1 * t)) * Float64(l / t)) / t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) * abs(k)) / Float64(l * l)) * Float64(tan(abs(k)) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 3.7e-40)
		tmp = (1.0 / abs(k)) * (((l / (t_1 * t)) * (l / t)) / t);
	else
		tmp = 2.0 / ((((abs(k) * t) * abs(k)) / (l * l)) * (tan(abs(k)) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.7e-40], N[(N[(1.0 / N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.69999999999999998e-40

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]

   if 3.69999999999999998e-40 < k

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \color{blue}{\ell}\right) \cdot \cos k}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(\frac{\sin k}{\cos k} \cdot \color{blue}{\sin k}\right)} \]

   6. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      10. lower-*.f64 75.4%

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   8. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      4. frac-times N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      8. lower-*.f64 64.1%

         \[\leadsto \frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

   10. Applied rewrites 64.1%

       \[\leadsto \frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.6% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 68000000:\\ \;\;\;\;\frac{2}{\left(\left|t\right| \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \frac{k}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 68000000.0)
    (/ 2.0 (* (* (fabs t) (* (tan k) (sin k))) (* k (/ k (* l l)))))
    (* (/ 1.0 k) (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 68000000.0) {
		tmp = 2.0 / ((fabs(t) * (tan(k) * sin(k))) * (k * (k / (l * l))));
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 68000000.0) {
		tmp = 2.0 / ((Math.abs(t) * (Math.tan(k) * Math.sin(k))) * (k * (k / (l * l))));
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 68000000.0:
		tmp = 2.0 / ((math.fabs(t) * (math.tan(k) * math.sin(k))) * (k * (k / (l * l))))
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 68000000.0)
		tmp = Float64(2.0 / Float64(Float64(abs(t) * Float64(tan(k) * sin(k))) * Float64(k * Float64(k / Float64(l * l)))));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 68000000.0)
		tmp = 2.0 / ((abs(t) * (tan(k) * sin(k))) * (k * (k / (l * l))));
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 68000000.0], N[(2.0 / N[(N[(N[Abs[t], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.8e7

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell \cdot \ell} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{\ell \cdot \ell}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{\ell \cdot \ell}}} \]

   6. Applied rewrites 62.7%

      \[\leadsto \frac{2}{\left(t \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \color{blue}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right)}} \]

   if 6.8e7 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.6% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 68000000:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot \frac{\left|t\right|}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 68000000.0)
    (/ 2.0 (* (* k (* k (/ (fabs t) (* l l)))) (* (tan k) (sin k))))
    (* (/ 1.0 k) (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 68000000.0) {
		tmp = 2.0 / ((k * (k * (fabs(t) / (l * l)))) * (tan(k) * sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 68000000.0) {
		tmp = 2.0 / ((k * (k * (Math.abs(t) / (l * l)))) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 68000000.0:
		tmp = 2.0 / ((k * (k * (math.fabs(t) / (l * l)))) * (math.tan(k) * math.sin(k)))
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 68000000.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * Float64(abs(t) / Float64(l * l)))) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 68000000.0)
		tmp = 2.0 / ((k * (k * (abs(t) / (l * l)))) * (tan(k) * sin(k)));
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 68000000.0], N[(2.0 / N[(N[(k * N[(k * N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.8e7

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \color{blue}{\ell}\right) \cdot \cos k}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(\frac{\sin k}{\cos k} \cdot \color{blue}{\sin k}\right)} \]

   6. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      8. lower-/.f64 64.5%

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   8. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

   if 6.8e7 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.2% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 68000000:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{\left|t\right|}{\ell \cdot \ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (copysign 1.0 t)
  (if (<= (fabs t) 68000000.0)
    (/ 2.0 (* k (* k (* (* (tan k) (sin k)) (/ (fabs t) (* l l))))))
    (* (/ 1.0 k) (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 68000000.0) {
		tmp = 2.0 / (k * (k * ((tan(k) * sin(k)) * (fabs(t) / (l * l)))));
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 68000000.0) {
		tmp = 2.0 / (k * (k * ((Math.tan(k) * Math.sin(k)) * (Math.abs(t) / (l * l)))));
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 68000000.0:
		tmp = 2.0 / (k * (k * ((math.tan(k) * math.sin(k)) * (math.fabs(t) / (l * l)))))
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 68000000.0)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * sin(k)) * Float64(abs(t) / Float64(l * l))))));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 68000000.0)
		tmp = 2.0 / (k * (k * ((tan(k) * sin(k)) * (abs(t) / (l * l)))));
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 68000000.0], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.8e7

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \color{blue}{\cos k}}\right)} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \cos \color{blue}{k}}\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \frac{{\sin k}^{2} \cdot t}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}\right)} \]

      16. times-frac N/A

          \[\leadsto \frac{2}{k \cdot \left(k \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

   6. Applied rewrites 64.5%

      \[\leadsto \frac{2}{k \cdot \color{blue}{\left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}} \]

   if 6.8e7 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 74.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t\_1}{\ell}\right) \cdot {k}^{2}}\\ \mathbf{elif}\;\left|t\right| \leq 8 \cdot 10^{+120}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \left|t\right|} \cdot \frac{\ell}{\left|t\right|}}{\left|t\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.5e-46)
      (/ 2.0 (* (* (/ k l) (/ t_1 l)) (pow k 2.0)))
      (if (<= (fabs t) 8e+120)
        (* (/ l (* (* (fabs t) (fabs t)) k)) (/ l t_1))
        (*
         (/ 1.0 k)
         (/ (* (/ l (* (sin k) (fabs t))) (/ l (fabs t))) (fabs t))))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1.5e-46) {
		tmp = 2.0 / (((k / l) * (t_1 / l)) * pow(k, 2.0));
	} else if (fabs(t) <= 8e+120) {
		tmp = (l / ((fabs(t) * fabs(t)) * k)) * (l / t_1);
	} else {
		tmp = (1.0 / k) * (((l / (sin(k) * fabs(t))) * (l / fabs(t))) / fabs(t));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.5e-46) {
		tmp = 2.0 / (((k / l) * (t_1 / l)) * Math.pow(k, 2.0));
	} else if (Math.abs(t) <= 8e+120) {
		tmp = (l / ((Math.abs(t) * Math.abs(t)) * k)) * (l / t_1);
	} else {
		tmp = (1.0 / k) * (((l / (Math.sin(k) * Math.abs(t))) * (l / Math.abs(t))) / Math.abs(t));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.5e-46:
		tmp = 2.0 / (((k / l) * (t_1 / l)) * math.pow(k, 2.0))
	elif math.fabs(t) <= 8e+120:
		tmp = (l / ((math.fabs(t) * math.fabs(t)) * k)) * (l / t_1)
	else:
		tmp = (1.0 / k) * (((l / (math.sin(k) * math.fabs(t))) * (l / math.fabs(t))) / math.fabs(t))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.5e-46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t_1 / l)) * (k ^ 2.0)));
	elseif (abs(t) <= 8e+120)
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) * Float64(l / t_1));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(Float64(Float64(l / Float64(sin(k) * abs(t))) * Float64(l / abs(t))) / abs(t)));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.5e-46)
		tmp = 2.0 / (((k / l) * (t_1 / l)) * (k ^ 2.0));
	elseif (abs(t) <= 8e+120)
		tmp = (l / ((abs(t) * abs(t)) * k)) * (l / t_1);
	else
		tmp = (1.0 / k) * (((l / (sin(k) * abs(t))) * (l / abs(t))) / abs(t));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.5e-46], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 8e+120], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999994e-46

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \color{blue}{\ell}\right) \cdot \cos k}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(\frac{\sin k}{\cos k} \cdot \color{blue}{\sin k}\right)} \]

   6. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      10. lower-*.f64 75.4%

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   8. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 59.1%

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{2}} \]

   11. Applied rewrites 59.1%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]

   if 1.49999999999999994e-46 < t < 7.9999999999999998e120

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k \cdot t} \]

      8. lower-/.f64 63.9%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 7.9999999999999998e120 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2 \cdot 1}}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \frac{1}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 53.2%

         \[\leadsto \frac{1}{\color{blue}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{1}{k}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{\sin k \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{\left(t \cdot t\right) \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot t}}}{t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k}}{t \cdot t}}{t} \]

      9. associate-/l* N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      11. lower-/.f64 58.8%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell \cdot \color{blue}{\frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{1}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}{t}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell \cdot \frac{\ell}{\sin k}}{t \cdot t}}}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}{t \cdot t}}{t} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}{t \cdot t}}{t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\frac{\ell}{\sin k} \cdot \ell}{\color{blue}{t \cdot t}}}{t} \]

      5. times-frac N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{\sin k}}{t} \cdot \frac{\ell}{t}}}{t} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{t} \cdot \frac{\ell}{t}}{t} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\color{blue}{\sin k \cdot t}} \cdot \frac{\ell}{t}}{t} \]

      11. lower-/.f64 64.7%

          \[\leadsto \frac{1}{k} \cdot \frac{\frac{\ell}{\sin k \cdot t} \cdot \color{blue}{\frac{\ell}{t}}}{t} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{1}{k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \frac{\ell}{t}}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 73.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t\_1}{\ell}\right) \cdot {k}^{2}}\\ \mathbf{elif}\;\left|t\right| \leq 10^{+144}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.5e-46)
      (/ 2.0 (* (* (/ k l) (/ t_1 l)) (pow k 2.0)))
      (if (<= (fabs t) 1e+144)
        (* (/ l (* (* (fabs t) (fabs t)) k)) (/ l t_1))
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1.5e-46) {
		tmp = 2.0 / (((k / l) * (t_1 / l)) * pow(k, 2.0));
	} else if (fabs(t) <= 1e+144) {
		tmp = (l / ((fabs(t) * fabs(t)) * k)) * (l / t_1);
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.5e-46) {
		tmp = 2.0 / (((k / l) * (t_1 / l)) * Math.pow(k, 2.0));
	} else if (Math.abs(t) <= 1e+144) {
		tmp = (l / ((Math.abs(t) * Math.abs(t)) * k)) * (l / t_1);
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.5e-46:
		tmp = 2.0 / (((k / l) * (t_1 / l)) * math.pow(k, 2.0))
	elif math.fabs(t) <= 1e+144:
		tmp = (l / ((math.fabs(t) * math.fabs(t)) * k)) * (l / t_1)
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.5e-46)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t_1 / l)) * (k ^ 2.0)));
	elseif (abs(t) <= 1e+144)
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) * Float64(l / t_1));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.5e-46)
		tmp = 2.0 / (((k / l) * (t_1 / l)) * (k ^ 2.0));
	elseif (abs(t) <= 1e+144)
		tmp = (l / ((abs(t) * abs(t)) * k)) * (l / t_1);
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.5e-46], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1e+144], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999994e-46

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \color{blue}{\ell}\right) \cdot \cos k}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(\frac{\sin k}{\cos k} \cdot \color{blue}{\sin k}\right)} \]

   6. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      10. lower-*.f64 75.4%

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   8. Applied rewrites 75.4%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 59.1%

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{2}} \]

   11. Applied rewrites 59.1%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]

   if 1.49999999999999994e-46 < t < 1.00000000000000002e144

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k \cdot t} \]

      8. lower-/.f64 63.9%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 1.00000000000000002e144 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lower-*.f64 65.1%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 70.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot \frac{\left|t\right|}{\ell \cdot \ell}\right)\right) \cdot {k}^{2}}\\ \mathbf{elif}\;\left|t\right| \leq 10^{+144}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1.5e-46)
      (/ 2.0 (* (* k (* k (/ (fabs t) (* l l)))) (pow k 2.0)))
      (if (<= (fabs t) 1e+144)
        (* (/ l (* (* (fabs t) (fabs t)) k)) (/ l t_1))
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1.5e-46) {
		tmp = 2.0 / ((k * (k * (fabs(t) / (l * l)))) * pow(k, 2.0));
	} else if (fabs(t) <= 1e+144) {
		tmp = (l / ((fabs(t) * fabs(t)) * k)) * (l / t_1);
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.5e-46) {
		tmp = 2.0 / ((k * (k * (Math.abs(t) / (l * l)))) * Math.pow(k, 2.0));
	} else if (Math.abs(t) <= 1e+144) {
		tmp = (l / ((Math.abs(t) * Math.abs(t)) * k)) * (l / t_1);
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.5e-46:
		tmp = 2.0 / ((k * (k * (math.fabs(t) / (l * l)))) * math.pow(k, 2.0))
	elif math.fabs(t) <= 1e+144:
		tmp = (l / ((math.fabs(t) * math.fabs(t)) * k)) * (l / t_1)
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.5e-46)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k * Float64(abs(t) / Float64(l * l)))) * (k ^ 2.0)));
	elseif (abs(t) <= 1e+144)
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) * Float64(l / t_1));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.5e-46)
		tmp = 2.0 / ((k * (k * (abs(t) / (l * l)))) * (k ^ 2.0));
	elseif (abs(t) <= 1e+144)
		tmp = (l / ((abs(t) * abs(t)) * k)) * (l / t_1);
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.5e-46], N[(2.0 / N[(N[(k * N[(k * N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1e+144], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999994e-46

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. 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. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \color{blue}{\ell}\right) \cdot \cos k}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos \color{blue}{k}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{\sin k \cdot \sin k}{\cos \color{blue}{k}}} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\ell \cdot \ell} \cdot \left(\frac{\sin k}{\cos k} \cdot \color{blue}{\sin k}\right)} \]

   6. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(\tan \color{blue}{k} \cdot \sin k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

      8. lower-/.f64 64.5%

         \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\tan k \cdot \sin k\right)} \]

   8. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot \left(\color{blue}{\tan k} \cdot \sin k\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot {k}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 55.4%

          \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot {k}^{2}} \]

   11. Applied rewrites 55.4%

       \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right) \cdot {k}^{\color{blue}{2}}} \]

   if 1.49999999999999994e-46 < t < 1.00000000000000002e144

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k \cdot t} \]

      8. lower-/.f64 63.9%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 1.00000000000000002e144 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lower-*.f64 65.1%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 69.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 6.5 \cdot 10^{-132}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{elif}\;\left|t\right| \leq 10^{+144}:\\ \;\;\;\;\frac{\ell}{\left(\left|t\right| \cdot \left|t\right|\right) \cdot k} \cdot \frac{\ell}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 6.5e-132)
      (* l (/ l (* (* (* (* k k) (fabs t)) (fabs t)) (fabs t))))
      (if (<= (fabs t) 1e+144)
        (* (/ l (* (* (fabs t) (fabs t)) k)) (/ l t_1))
        (* (/ l (* (* t_1 (fabs t)) t_1)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 6.5e-132) {
		tmp = l * (l / ((((k * k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else if (fabs(t) <= 1e+144) {
		tmp = (l / ((fabs(t) * fabs(t)) * k)) * (l / t_1);
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 6.5e-132) {
		tmp = l * (l / ((((k * k) * Math.abs(t)) * Math.abs(t)) * Math.abs(t)));
	} else if (Math.abs(t) <= 1e+144) {
		tmp = (l / ((Math.abs(t) * Math.abs(t)) * k)) * (l / t_1);
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 6.5e-132:
		tmp = l * (l / ((((k * k) * math.fabs(t)) * math.fabs(t)) * math.fabs(t)))
	elif math.fabs(t) <= 1e+144:
		tmp = (l / ((math.fabs(t) * math.fabs(t)) * k)) * (l / t_1)
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 6.5e-132)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(Float64(k * k) * abs(t)) * abs(t)) * abs(t))));
	elseif (abs(t) <= 1e+144)
		tmp = Float64(Float64(l / Float64(Float64(abs(t) * abs(t)) * k)) * Float64(l / t_1));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 6.5e-132)
		tmp = l * (l / ((((k * k) * abs(t)) * abs(t)) * abs(t)));
	elseif (abs(t) <= 1e+144)
		tmp = (l / ((abs(t) * abs(t)) * k)) * (l / t_1);
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 6.5e-132], N[(l * N[(l / N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1e+144], N[(N[(l / N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.49999999999999991e-132

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. lower-*.f64 61.5%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t} \]

   8. Applied rewrites 61.5%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

   if 6.49999999999999991e-132 < t < 1.00000000000000002e144

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

      5. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\color{blue}{\ell}}{k \cdot t} \]

      8. lower-/.f64 63.9%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   10. Applied rewrites 63.9%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 1.00000000000000002e144 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lower-*.f64 65.1%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 69.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot t\right) \cdot t\_1} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(\left|k\right| \cdot \left|k\right|\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (fabs k) t)))
   (if (<= (fabs k) 1.1e-156)
     (* (/ l (* (* t_1 t) t_1)) l)
     (* (/ l (* (* (* (fabs k) (fabs k)) t) t)) (/ l t)))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 1.1e-156) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else {
		tmp = (l / (((fabs(k) * fabs(k)) * t) * t)) * (l / t);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 1.1d-156) then
        tmp = (l / ((t_1 * t) * t_1)) * l
    else
        tmp = (l / (((abs(k) * abs(k)) * t) * t)) * (l / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 1.1e-156) {
		tmp = (l / ((t_1 * t) * t_1)) * l;
	} else {
		tmp = (l / (((Math.abs(k) * Math.abs(k)) * t) * t)) * (l / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 1.1e-156:
		tmp = (l / ((t_1 * t) * t_1)) * l
	else:
		tmp = (l / (((math.fabs(k) * math.fabs(k)) * t) * t)) * (l / t)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 1.1e-156)
		tmp = Float64(Float64(l / Float64(Float64(t_1 * t) * t_1)) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(abs(k) * abs(k)) * t) * t)) * Float64(l / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 1.1e-156)
		tmp = (l / ((t_1 * t) * t_1)) * l;
	else
		tmp = (l / (((abs(k) * abs(k)) * t) * t)) * (l / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.1e-156], N[(N[(l / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1e-156

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lower-*.f64 65.1%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 1.1e-156 < k

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      7. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{t} \]

      11. lower-/.f64 62.8%

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{\color{blue}{t}} \]

   8. Applied rewrites 62.8%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 67.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 10^{+47}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot \left|t\right|\right) \cdot \left|t\right|\right) \cdot \left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t\_1 \cdot \left|t\right|\right) \cdot t\_1} \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (fabs t))))
   (*
    (copysign 1.0 t)
    (if (<= (fabs t) 1e+47)
      (* l (/ l (* (* (* (* k k) (fabs t)) (fabs t)) (fabs t))))
      (* (/ l (* (* t_1 (fabs t)) t_1)) l)))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1e+47) {
		tmp = l * (l / ((((k * k) * fabs(t)) * fabs(t)) * fabs(t)));
	} else {
		tmp = (l / ((t_1 * fabs(t)) * t_1)) * l;
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1e+47) {
		tmp = l * (l / ((((k * k) * Math.abs(t)) * Math.abs(t)) * Math.abs(t)));
	} else {
		tmp = (l / ((t_1 * Math.abs(t)) * t_1)) * l;
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1e+47:
		tmp = l * (l / ((((k * k) * math.fabs(t)) * math.fabs(t)) * math.fabs(t)))
	else:
		tmp = (l / ((t_1 * math.fabs(t)) * t_1)) * l
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1e+47)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(Float64(k * k) * abs(t)) * abs(t)) * abs(t))));
	else
		tmp = Float64(Float64(l / Float64(Float64(t_1 * abs(t)) * t_1)) * l);
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1e+47)
		tmp = l * (l / ((((k * k) * abs(t)) * abs(t)) * abs(t)));
	else
		tmp = (l / ((t_1 * abs(t)) * t_1)) * l;
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1e+47], N[(l * N[(l / N[(N[(N[(N[(k * k), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$1 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1e47

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. lower-*.f64 61.5%

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t} \]

   8. Applied rewrites 61.5%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

   if 1e47 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 50.4%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 54.8%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. unpow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lower-*.f64 58.0%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.0%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.0%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.1%

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lower-*.f64 65.1%

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 65.1% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* k t) t) (* k t))) l))

C

double code(double t, double l, double k) {
	return (l / (((k * t) * t) * (k * t))) * l;
}

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 = (l / (((k * t) * t) * (k * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * t) * t) * (k * t))) * l;
}

Python

def code(t, l, k):
	return (l / (((k * t) * t) * (k * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * t) * t) * Float64(k * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * t) * t) * (k * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * t), $MachinePrecision] * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.4%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 54.8%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. cube-mult N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   13. lower-*.f64 58.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

   15. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   16. lower-*.f64 58.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

6. Applied rewrites 58.0%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 58.0%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 62.1%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.1%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   6. lower-*.f64 65.1%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 65.1%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 22: 61.0% accurate, 6.6× speedup

Math

\[\frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (* (* (* k t) k) (* t t))) l))

C

double code(double t, double l, double k) {
	return (l / (((k * t) * k) * (t * t))) * l;
}

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 = (l / (((k * t) * k) * (t * t))) * l
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * t) * k) * (t * t))) * l;
}

Python

def code(t, l, k):
	return (l / (((k * t) * k) * (t * t))) * l

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * t) * k) * Float64(t * t))) * l)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * t) * k) * (t * t))) * l;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 50.4%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 54.8%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. cube-mult N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   13. lower-*.f64 58.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

   15. unpow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   16. lower-*.f64 58.0%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

6. Applied rewrites 58.0%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 58.0%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 62.1%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.1%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   7. lower-*.f64 61.0%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

10. Applied rewrites 61.0%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(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))))