Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.9% → 91.3%

Time: 7.8s

Alternatives: 14

Speedup: 5.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-158}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \mathbf{elif}\;k\_m \leq 1.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k\_m, k\_m \cdot \left(\frac{\sin k\_m}{\ell} \cdot \tan k\_m\right), \left(2 \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1e-158)
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)
   (if (<= k_m 1.7e+102)
     (/
      2.0
      (/
       (*
        t
        (fma
         k_m
         (* k_m (* (/ (sin k_m) l) (tan k_m)))
         (* (* 2.0 (* (/ t l) t)) (* (tan k_m) (sin k_m)))))
       l))
     (*
      (*
       (* (cos k_m) 2.0)
       (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
      (/ l k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1e-158) {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	} else if (k_m <= 1.7e+102) {
		tmp = 2.0 / ((t * fma(k_m, (k_m * ((sin(k_m) / l) * tan(k_m))), ((2.0 * ((t / l) * t)) * (tan(k_m) * sin(k_m))))) / l);
	} else {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1e-158)
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	elseif (k_m <= 1.7e+102)
		tmp = Float64(2.0 / Float64(Float64(t * fma(k_m, Float64(k_m * Float64(Float64(sin(k_m) / l) * tan(k_m))), Float64(Float64(2.0 * Float64(Float64(t / l) * t)) * Float64(tan(k_m) * sin(k_m))))) / l));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1e-158], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[k$95$m, 1.7e+102], N[(2.0 / N[(N[(t * N[(k$95$m * N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.00000000000000006e-158

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

   if 1.00000000000000006e-158 < k < 1.7e102

   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

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 34.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 34.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 74.8%

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

   10. Applied rewrites 81.7%

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

   if 1.7e102 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 70.0%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 44:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k\_m \cdot \frac{t}{\ell}\right) \cdot \left(\sin k\_m \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 44.0)
   (*
    (* (* (cos k_m) 2.0) (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
    (/ l k_m))
   (/
    2.0
    (*
     (* (* (tan k_m) (/ t l)) (* (sin k_m) (* (/ t l) t)))
     (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 44.0) {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	} else {
		tmp = 2.0 / (((tan(k_m) * (t / l)) * (sin(k_m) * ((t / l) * t))) * ((1.0 + pow((k_m / t), 2.0)) + 1.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 44.0)
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k_m) * Float64(t / l)) * Float64(sin(k_m) * Float64(Float64(t / l) * t))) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 44.0], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 44

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 70.0%

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

   if 44 < 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

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 34.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 34.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. unpow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.0

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

   7. Applied rewrites 75.0%

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

Alternative 3: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(\mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right) \cdot \tan k\_m\right) \cdot \left(\sin k\_m \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6e+22)
   (*
    (* (* (cos k_m) 2.0) (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
    (/ l k_m))
   (/
    2.0
    (/
     (*
      (*
       (* (* (fma (/ k_m (* t t)) k_m 2.0) (tan k_m)) (* (sin k_m) t))
       (/ t l))
      t)
     l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6e+22) {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	} else {
		tmp = 2.0 / (((((fma((k_m / (t * t)), k_m, 2.0) * tan(k_m)) * (sin(k_m) * t)) * (t / l)) * t) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6e+22)
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(k_m / Float64(t * t)), k_m, 2.0) * tan(k_m)) * Float64(sin(k_m) * t)) * Float64(t / l)) * t) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6e+22], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6e22

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 70.0%

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

   if 6e22 < 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

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 34.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 34.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. unpow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 64.6%

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

Alternative 4: 78.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(\left(\mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right) \cdot \tan k\_m\right) \cdot \left(\sin k\_m \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6e+22)
   (*
    (* (* (cos k_m) 2.0) (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
    (/ l k_m))
   (/
    (* 2.0 l)
    (*
     (*
      (* (* (fma (/ k_m (* t t)) k_m 2.0) (tan k_m)) (* (sin k_m) t))
      (/ t l))
     t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6e+22) {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	} else {
		tmp = (2.0 * l) / ((((fma((k_m / (t * t)), k_m, 2.0) * tan(k_m)) * (sin(k_m) * t)) * (t / l)) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6e+22)
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(Float64(Float64(Float64(fma(Float64(k_m / Float64(t * t)), k_m, 2.0) * tan(k_m)) * Float64(sin(k_m) * t)) * Float64(t / l)) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6e+22], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[(N[(N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6e22

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 70.0%

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

   if 6e22 < 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

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 34.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 34.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. unpow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 64.7%

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

Alternative 5: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right) \cdot \left(\left(\left(\sin k\_m \cdot t\right) \cdot \left(\frac{t}{\ell \cdot \ell} \cdot t\right)\right) \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+74)
   (/
    2.0
    (*
     (fma k_m (/ k_m (* t t)) 2.0)
     (* (* (* (sin k_m) t) (* (/ t (* l l)) t)) (tan k_m))))
   (*
    (* (* (cos k_m) 2.0) (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
    (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+74) {
		tmp = 2.0 / (fma(k_m, (k_m / (t * t)), 2.0) * (((sin(k_m) * t) * ((t / (l * l)) * t)) * tan(k_m)));
	} else {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+74)
		tmp = Float64(2.0 / Float64(fma(k_m, Float64(k_m / Float64(t * t)), 2.0) * Float64(Float64(Float64(sin(k_m) * t) * Float64(Float64(t / Float64(l * l)) * t)) * tan(k_m))));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+74], N[(2.0 / N[(N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999963e74

   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

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 34.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 34.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 56.2%

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

   if 4.99999999999999963e74 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 70.0%

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

Alternative 6: 78.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+62}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m}\right) \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.5e+62)
   (*
    (* (* (cos k_m) 2.0) (/ l (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m)))
    (/ l k_m))
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.5e+62) {
		tmp = ((cos(k_m) * 2.0) * (l / ((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m))) * (l / k_m);
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.5e+62)
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * Float64(l / Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m))) * Float64(l / k_m));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.5e+62], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.49999999999999984e62

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied rewrites 70.0%

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

   if 3.49999999999999984e62 < 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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

Alternative 7: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot 2\right) \cdot \ell\right) \cdot \frac{\ell}{\left(\left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.2e+27)
   (*
    (* (* (cos k_m) 2.0) l)
    (/ l (* (* (* (fma (cos (+ k_m k_m)) -0.5 0.5) t) k_m) k_m)))
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.2e+27) {
		tmp = ((cos(k_m) * 2.0) * l) * (l / (((fma(cos((k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m));
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.2e+27)
		tmp = Float64(Float64(Float64(cos(k_m) * 2.0) * l) * Float64(l / Float64(Float64(Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t) * k_m) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.2e+27], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(l / N[(N[(N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1999999999999999e27

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 66.2

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

   8. Applied rewrites 66.2%

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

   if 2.1999999999999999e27 < 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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

Alternative 8: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0022:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k\_m \cdot \frac{\ell \cdot \ell}{\left(\left(k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right) \cdot t\right) \cdot k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0022)
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)
   (*
    2.0
    (*
     (cos k_m)
     (/ (* l l) (* (* (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5)) t) k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0022) {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	} else {
		tmp = 2.0 * (cos(k_m) * ((l * l) / (((k_m * fma(cos((k_m + k_m)), -0.5, 0.5)) * t) * k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0022)
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	else
		tmp = Float64(2.0 * Float64(cos(k_m) * Float64(Float64(l * l) / Float64(Float64(Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)) * t) * k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0022], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00220000000000000013

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

   if 0.00220000000000000013 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 59.8

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval 59.8

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

   8. Applied rewrites 59.8%

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

Alternative 9: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0022:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\ell \cdot \ell} \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0022)
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)
   (/ 2.0 (* (/ (* (* k_m k_m) t) (* l l)) (* (tan k_m) (sin k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0022) {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (tan(k_m) * sin(k_m)));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 0.0022d0) then
        tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t) / (l * l)) * (tan(k_m) * sin(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0022) {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (Math.tan(k_m) * Math.sin(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 0.0022:
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
	else:
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (math.tan(k_m) * math.sin(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0022)
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / Float64(l * l)) * Float64(tan(k_m) * sin(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 0.0022)
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	else
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (tan(k_m) * sin(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0022], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00220000000000000013

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

   if 0.00220000000000000013 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 60.3

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

   4. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

   6. Applied rewrites 61.7%

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

Alternative 10: 68.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{1.5}}}{{t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.3e+91)
   (/ (/ (* (/ l k_m) (/ l k_m)) (pow t 1.5)) (pow t 1.5))
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.3e+91) {
		tmp = (((l / k_m) * (l / k_m)) / pow(t, 1.5)) / pow(t, 1.5);
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 3.3d+91) then
        tmp = (((l / k_m) * (l / k_m)) / (t ** 1.5d0)) / (t ** 1.5d0)
    else
        tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.3e+91) {
		tmp = (((l / k_m) * (l / k_m)) / Math.pow(t, 1.5)) / Math.pow(t, 1.5);
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3.3e+91:
		tmp = (((l / k_m) * (l / k_m)) / math.pow(t, 1.5)) / math.pow(t, 1.5)
	else:
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.3e+91)
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 1.5)) / (t ^ 1.5));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3.3e+91)
		tmp = (((l / k_m) * (l / k_m)) / (t ^ 1.5)) / (t ^ 1.5);
	else
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.3e+91], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.30000000000000017e91

   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 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 29.9

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{{t}^{1.5}}}{{t}^{1.5}} \]

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 29.9

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{1.5}}}{{t}^{1.5}} \]

   6. Applied rewrites 29.9%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{1.5}}}{\color{blue}{{t}^{1.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 32.8

         \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5}}}{{t}^{1.5}} \]

   8. Applied rewrites 32.8%

      \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5}}}{{t}^{1.5}} \]

   if 3.30000000000000017e91 < 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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

Alternative 11: 65.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.58 \cdot 10^{+91}:\\ \;\;\;\;{t}^{-1.5} \cdot \left({t}^{-1.5} \cdot \left(\frac{\frac{\ell}{k\_m}}{k\_m} \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.58e+91)
   (* (pow t -1.5) (* (pow t -1.5) (* (/ (/ l k_m) k_m) l)))
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.58e+91) {
		tmp = pow(t, -1.5) * (pow(t, -1.5) * (((l / k_m) / k_m) * l));
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.58d+91) then
        tmp = (t ** (-1.5d0)) * ((t ** (-1.5d0)) * (((l / k_m) / k_m) * l))
    else
        tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.58e+91) {
		tmp = Math.pow(t, -1.5) * (Math.pow(t, -1.5) * (((l / k_m) / k_m) * l));
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.58e+91:
		tmp = math.pow(t, -1.5) * (math.pow(t, -1.5) * (((l / k_m) / k_m) * l))
	else:
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.58e+91)
		tmp = Float64((t ^ -1.5) * Float64((t ^ -1.5) * Float64(Float64(Float64(l / k_m) / k_m) * l)));
	else
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.58e+91)
		tmp = (t ^ -1.5) * ((t ^ -1.5) * (((l / k_m) / k_m) * l));
	else
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.58e+91], N[(N[Power[t, -1.5], $MachinePrecision] * N[(N[Power[t, -1.5], $MachinePrecision] * N[(N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5799999999999999e91

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l/ N/A

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

      17. lift-/.f64 N/A

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

      18. mult-flip N/A

          \[\leadsto \frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{\frac{3}{2}}} \cdot \color{blue}{\frac{1}{{t}^{\frac{3}{2}}}} \]

   8. Applied rewrites 29.7%

      \[\leadsto {t}^{-1.5} \cdot \color{blue}{\left({t}^{-1.5} \cdot \left(\frac{\ell}{k \cdot k} \cdot \ell\right)\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 31.9

         \[\leadsto {t}^{-1.5} \cdot \left({t}^{-1.5} \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \ell\right)\right) \]

   10. Applied rewrites 31.9%

       \[\leadsto {t}^{-1.5} \cdot \left({t}^{-1.5} \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \ell\right)\right) \]

   if 1.5799999999999999e91 < 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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

Alternative 12: 64.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{-158}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.95e-158)
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)
   (* l (/ (/ (/ l (* (* k_m k_m) t)) t) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e-158) {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	} else {
		tmp = l * (((l / ((k_m * k_m) * t)) / t) / t);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.95d-158) then
        tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
    else
        tmp = l * (((l / ((k_m * k_m) * t)) / t) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e-158) {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	} else {
		tmp = l * (((l / ((k_m * k_m) * t)) / t) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.95e-158:
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
	else:
		tmp = l * (((l / ((k_m * k_m) * t)) / t) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e-158)
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(Float64(k_m * k_m) * t)) / t) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.95e-158)
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	else
		tmp = l * (((l / ((k_m * k_m) * t)) / t) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.95e-158], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(l * N[(N[(N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9499999999999998e-158

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

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

   10. Applied rewrites 65.1%

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

   if 1.9499999999999998e-158 < 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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 62.7

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

   8. Applied rewrites 62.7%

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

Alternative 13: 34.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k\_m}}{\left(t \cdot t\right) \cdot k\_m} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 8e+152)
   (* (/ (/ l (* t k_m)) (* (* t t) k_m)) l)
   (* (/ l (* t (* (* t k_m) (* t k_m)))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8e+152) {
		tmp = ((l / (t * k_m)) / ((t * t) * k_m)) * l;
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 8d+152) then
        tmp = ((l / (t * k_m)) / ((t * t) * k_m)) * l
    else
        tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8e+152) {
		tmp = ((l / (t * k_m)) / ((t * t) * k_m)) * l;
	} else {
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 8e+152:
		tmp = ((l / (t * k_m)) / ((t * t) * k_m)) * l
	else:
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 8e+152)
		tmp = Float64(Float64(Float64(l / Float64(t * k_m)) / Float64(Float64(t * t) * k_m)) * l);
	else
		tmp = Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 8e+152)
		tmp = ((l / (t * k_m)) / ((t * t) * k_m)) * l;
	else
		tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 8e+152], N[(N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000004e152

   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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.7

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

   10. Applied rewrites 63.7%

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

   if 8.0000000000000004e152 < 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 50.1

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

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 54.7

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

      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 57.5

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

      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 57.5

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

   6. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.5

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

      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. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. 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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 65.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.1

          \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   10. Applied rewrites 65.1%

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 34.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{t \cdot \left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)} \cdot \ell \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* t (* (* t k_m) (* t k_m)))) l))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (t * ((t * k_m) * (t * k_m)))) * l;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l / (t * ((t * k_m) * (t * k_m)))) * l
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / (t * ((t * k_m) * (t * k_m)))) * l;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (t * ((t * k_m) * (t * k_m)))) * l

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(t * Float64(Float64(t * k_m) * Float64(t * k_m)))) * l)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / (t * ((t * k_m) * (t * k_m)))) * l;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(t * N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $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 50.1

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 54.7

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   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 57.5

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   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 57.5

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

6. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
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 57.5

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   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. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   13. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   14. 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. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   7. associate-*l* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   9. lower-*.f64 65.1

      \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 65.1

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   14. *-commutative N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   15. lower-*.f64 65.1

       \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

10. Applied rewrites 65.1%

    \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))