Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 79.3%

Time: 7.6s

Alternatives: 16

Speedup: 3.3×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \frac{1}{{l\_m}^{-2}}} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 4e+154)
   (/
    2.0
    (*
     (/
      (fma 2.0 (pow (* (sin k) t) 2.0) (pow (* (sin k) k) 2.0))
      (* (cos k) (/ 1.0 (pow l_m -2.0))))
     t))
   (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 4e+154) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (1.0 / pow(l_m, -2.0)))) * t);
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 4e+154)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(1.0 / (l_m ^ -2.0)))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 4e+154], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(1.0 / N[Power[l$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000015e154

   1. Initial program 59.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. metadata-eval N/A

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

      6. pow-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if 4.00000000000000015e154 < l

   1. Initial program 28.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 58.8

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

   7. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 72.3

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

   9. Applied rewrites 72.3%

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

Alternative 2: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 4e+154)
   (/
    2.0
    (*
     (/
      (fma 2.0 (pow (* (sin k) t) 2.0) (pow (* (sin k) k) 2.0))
      (* (cos k) (* l_m l_m)))
     t))
   (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 4e+154) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l_m * l_m))) * t);
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 4e+154)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 4e+154], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000015e154

   1. Initial program 59.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.9%

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

   if 4.00000000000000015e154 < l

   1. Initial program 28.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 58.8

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

   7. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 72.3

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

   9. Applied rewrites 72.3%

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

Alternative 3: 54.2% accurate, 0.9× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<=
      (*
       (* (* (/ (pow t 3.0) (* l_m l_m)) (sin k)) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))
      INFINITY)
   (/ (* l_m l_m) (* (* k k) (* (* t t) t)))
   (/
    2.0
    (* (/ (* (* k k) (* k k)) (* (fma -0.5 (* k k) 1.0) (* l_m l_m))) t))))

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 85.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if +inf.0 < (*.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 0.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 27.0

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 27.0%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 19.9

           \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 19.9%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 22.8

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

   14. Applied rewrites 22.8%

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(k \cdot 2\right), k \cdot k, {\left(\sin k \cdot t\right)}^{2} \cdot 2\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 4e+154)
   (/
    2.0
    (*
     (/
      (fma
       (- 0.5 (* 0.5 (cos (* k 2.0))))
       (* k k)
       (* (pow (* (sin k) t) 2.0) 2.0))
      (* (cos k) (* l_m l_m)))
     t))
   (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 4e+154) {
		tmp = 2.0 / ((fma((0.5 - (0.5 * cos((k * 2.0)))), (k * k), (pow((sin(k) * t), 2.0) * 2.0)) / (cos(k) * (l_m * l_m))) * t);
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 4e+154)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(k * 2.0)))), Float64(k * k), Float64((Float64(sin(k) * t) ^ 2.0) * 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 4e+154], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000015e154

   1. Initial program 59.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   7. Applied rewrites 81.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.1

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

   9. Applied rewrites 80.1%

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

   if 4.00000000000000015e154 < l

   1. Initial program 28.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 58.8

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

   7. Applied rewrites 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 72.3

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

   9. Applied rewrites 72.3%

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

Alternative 5: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \frac{1}{{l\_m}^{-2}}} \cdot t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{l\_m} \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 6e-72)
   (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (/ 1.0 (pow l_m -2.0)))) t))
   (if (<= t 1.35e+145)
     (/
      2.0
      (*
       (* (* (* (/ (* t t) l_m) (/ t l_m)) (sin k)) (tan k))
       (+ (pow (/ k t) 2.0) 2.0)))
     (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 6e-72) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / (cos(k) * (1.0 / pow(l_m, -2.0)))) * t);
	} else if (t <= 1.35e+145) {
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (pow((k / t), 2.0) + 2.0));
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 6d-72) then
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) / (cos(k) * (1.0d0 / (l_m ** (-2.0d0))))) * t)
    else if (t <= 1.35d+145) then
        tmp = 2.0d0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (((k / t) ** 2.0d0) + 2.0d0))
    else
        tmp = 2.0d0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 6e-72) {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) / (Math.cos(k) * (1.0 / Math.pow(l_m, -2.0)))) * t);
	} else if (t <= 1.35e+145) {
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * Math.sin(k)) * Math.tan(k)) * (Math.pow((k / t), 2.0) + 2.0));
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if t <= 6e-72:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) / (math.cos(k) * (1.0 / math.pow(l_m, -2.0)))) * t)
	elif t <= 1.35e+145:
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * math.sin(k)) * math.tan(k)) * (math.pow((k / t), 2.0) + 2.0))
	else:
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * math.sin(k)) * math.tan(k)) * 2.0)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 6e-72)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * Float64(1.0 / (l_m ^ -2.0)))) * t));
	elseif (t <= 1.35e+145)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l_m) * Float64(t / l_m)) * sin(k)) * tan(k)) * Float64((Float64(k / t) ^ 2.0) + 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (t <= 6e-72)
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) / (cos(k) * (1.0 / (l_m ^ -2.0)))) * t);
	elseif (t <= 1.35e+145)
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (((k / t) ^ 2.0) + 2.0));
	else
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 6e-72], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(1.0 / N[Power[l$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+145], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6e-72

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. metadata-eval N/A

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

      6. pow-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k \cdot \frac{1}{{\ell}^{-2}}} \cdot t} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot {k}^{2}}{\cos k \cdot \frac{1}{{\ell}^{-2}}} \cdot t} \]

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 67.3

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

   10. Applied rewrites 67.3%

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

   if 6e-72 < t < 1.35000000000000011e145

   1. Initial program 71.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 76.2

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

   10. Applied rewrites 76.2%

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

   if 1.35000000000000011e145 < t

   1. Initial program 65.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 66.9

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

   7. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 87.5

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

   9. Applied rewrites 87.5%

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

Alternative 6: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{l\_m} \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 6e-72)
   (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (* l_m l_m))) t))
   (if (<= t 1.35e+145)
     (/
      2.0
      (*
       (* (* (* (/ (* t t) l_m) (/ t l_m)) (sin k)) (tan k))
       (+ (pow (/ k t) 2.0) 2.0)))
     (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 6e-72) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / (cos(k) * (l_m * l_m))) * t);
	} else if (t <= 1.35e+145) {
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (pow((k / t), 2.0) + 2.0));
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 6d-72) then
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) / (cos(k) * (l_m * l_m))) * t)
    else if (t <= 1.35d+145) then
        tmp = 2.0d0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (((k / t) ** 2.0d0) + 2.0d0))
    else
        tmp = 2.0d0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 6e-72) {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) / (Math.cos(k) * (l_m * l_m))) * t);
	} else if (t <= 1.35e+145) {
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * Math.sin(k)) * Math.tan(k)) * (Math.pow((k / t), 2.0) + 2.0));
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if t <= 6e-72:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) / (math.cos(k) * (l_m * l_m))) * t)
	elif t <= 1.35e+145:
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * math.sin(k)) * math.tan(k)) * (math.pow((k / t), 2.0) + 2.0))
	else:
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * math.sin(k)) * math.tan(k)) * 2.0)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 6e-72)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t));
	elseif (t <= 1.35e+145)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l_m) * Float64(t / l_m)) * sin(k)) * tan(k)) * Float64((Float64(k / t) ^ 2.0) + 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (t <= 6e-72)
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) / (cos(k) * (l_m * l_m))) * t);
	elseif (t <= 1.35e+145)
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (((k / t) ^ 2.0) + 2.0));
	else
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 6e-72], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+145], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6e-72

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if 6e-72 < t < 1.35000000000000011e145

   1. Initial program 71.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 76.2

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

   10. Applied rewrites 76.2%

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

   if 1.35000000000000011e145 < t

   1. Initial program 65.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 66.9

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

   7. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 87.5

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

   9. Applied rewrites 87.5%

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

Alternative 7: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{l\_m} \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)}{t \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 6e-72)
   (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (* l_m l_m))) t))
   (if (<= t 4e+138)
     (/
      2.0
      (*
       (* (* (* (/ (* t t) l_m) (/ t l_m)) (sin k)) (tan k))
       (/ (fma (* t t) 2.0 (* k k)) (* t t))))
     (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 6e-72) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / (cos(k) * (l_m * l_m))) * t);
	} else if (t <= 4e+138) {
		tmp = 2.0 / ((((((t * t) / l_m) * (t / l_m)) * sin(k)) * tan(k)) * (fma((t * t), 2.0, (k * k)) / (t * t)));
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 6e-72)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t));
	elseif (t <= 4e+138)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l_m) * Float64(t / l_m)) * sin(k)) * tan(k)) * Float64(fma(Float64(t * t), 2.0, Float64(k * k)) / Float64(t * t))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 6e-72], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+138], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6e-72

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if 6e-72 < t < 4.0000000000000001e138

   1. Initial program 71.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 76.1

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

   10. Applied rewrites 76.1%

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

   if 4.0000000000000001e138 < t

   1. Initial program 65.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 66.9

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

   7. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 87.5

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

   9. Applied rewrites 87.5%

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

Alternative 8: 71.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8 \cdot 10^{+131}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{l\_m}\right) \cdot \frac{t}{l\_m}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 8e+131)
   (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l_m l_m))) t))
   (/ 2.0 (* (* (* (* (* t (/ t l_m)) (/ t l_m)) (sin k)) (tan k)) 2.0))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 8e+131) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t);
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l_m <= 8d+131) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l_m * l_m))) * t)
    else
        tmp = 2.0d0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 8e+131) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l_m * l_m))) * t);
	} else {
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	tmp = 0
	if l_m <= 8e+131:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l_m * l_m))) * t)
	else:
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * math.sin(k)) * math.tan(k)) * 2.0)
	return tmp

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 8e+131)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * Float64(t / l_m)) * Float64(t / l_m)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, k)
	tmp = 0.0;
	if (l_m <= 8e+131)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t);
	else
		tmp = 2.0 / (((((t * (t / l_m)) * (t / l_m)) * sin(k)) * tan(k)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 8e+131], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.9999999999999993e131

   1. Initial program 60.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

   if 7.9999999999999993e131 < l

   1. Initial program 26.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 57.1

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 69.6

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

   9. Applied rewrites 69.6%

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

Alternative 9: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l_m l_m))) t)))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t);
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l_m * l_m))) * t)
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l_m * l_m))) * t);
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l_m * l_m))) * t)

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 78.2%

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

6. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 70.1

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

8. Applied rewrites 70.1%

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

Alternative 10: 57.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 0.05:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 0.05)
   (/
    2.0
    (*
     (/
      (*
       (fma (fma -0.6666666666666666 (* t t) 1.0) (* k k) (* (* t t) 2.0))
       (* k k))
      (* (cos k) (* l_m l_m)))
     t))
   (/ 2.0 (* (* (/ (pow (* k t) 2.0) (* l_m l_m)) 2.0) t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 0.05) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / (cos(k) * (l_m * l_m))) * t);
	} else {
		tmp = 2.0 / (((pow((k * t), 2.0) / (l_m * l_m)) * 2.0) * t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 0.05)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / Float64(cos(k) * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 0.05], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.050000000000000003

   1. Initial program 52.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 53.9

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

   8. Applied rewrites 53.9%

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

   if 0.050000000000000003 < t

   1. Initial program 66.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 69.3

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

   8. Applied rewrites 69.3%

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

Alternative 11: 64.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.7 \cdot 10^{+174}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 1.7e+174)
   (/ 2.0 (* (* (/ (pow (* k t) 2.0) (* l_m l_m)) 2.0) t))
   (/
    2.0
    (*
     (/ (* (* (* k t) (* k t)) 2.0) (* (fma -0.5 (* k k) 1.0) (* l_m l_m)))
     t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (l_m <= 1.7e+174) {
		tmp = 2.0 / (((pow((k * t), 2.0) / (l_m * l_m)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((((k * t) * (k * t)) * 2.0) / (fma(-0.5, (k * k), 1.0) * (l_m * l_m))) * t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (l_m <= 1.7e+174)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0) / Float64(fma(-0.5, Float64(k * k), 1.0) * Float64(l_m * l_m))) * t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[l$95$m, 1.7e+174], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7000000000000001e174

   1. Initial program 60.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 70.0

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

   8. Applied rewrites 70.0%

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

   if 1.7000000000000001e174 < l

   1. Initial program 22.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.9

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 42.9%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 43.2

          \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 43.2%

       \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 43.2

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

   13. Applied rewrites 43.2%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{if}\;t \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (/
            (* (* (* k t) (* k t)) 2.0)
            (* (fma -0.5 (* k k) 1.0) (* l_m l_m)))
           t))))
   (if (<= t 2.9e-261)
     t_1
     (if (<= t 2.25e+130)
       (/
        2.0
        (*
         (/
          (*
           (fma (fma -0.6666666666666666 (* t t) 1.0) (* k k) (* (* t t) 2.0))
           (* k k))
          (* 1.0 (* l_m l_m)))
         t))
       (if (<= t 1.35e+227) t_1 (* l_m (/ l_m (* (* k k) (pow t 3.0)))))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double t_1 = 2.0 / (((((k * t) * (k * t)) * 2.0) / (fma(-0.5, (k * k), 1.0) * (l_m * l_m))) * t);
	double tmp;
	if (t <= 2.9e-261) {
		tmp = t_1;
	} else if (t <= 2.25e+130) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / (1.0 * (l_m * l_m))) * t);
	} else if (t <= 1.35e+227) {
		tmp = t_1;
	} else {
		tmp = l_m * (l_m / ((k * k) * pow(t, 3.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	t_1 = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0) / Float64(fma(-0.5, Float64(k * k), 1.0) * Float64(l_m * l_m))) * t))
	tmp = 0.0
	if (t <= 2.9e-261)
		tmp = t_1;
	elseif (t <= 2.25e+130)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_m))) * t));
	elseif (t <= 1.35e+227)
		tmp = t_1;
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * k) * (t ^ 3.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 2.9e-261], t$95$1, If[LessEqual[t, 2.25e+130], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+227], t$95$1, N[(l$95$m * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.89999999999999985e-261 or 2.2500000000000002e130 < t < 1.3499999999999999e227

   1. Initial program 53.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 54.5

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 54.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 53.6

          \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 53.6%

       \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 53.6

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

   13. Applied rewrites 53.6%

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

   if 2.89999999999999985e-261 < t < 2.2500000000000002e130

   1. Initial program 56.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 46.8

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 46.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 52.7

           \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 52.7%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

   14. Applied rewrites 67.7%

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

   if 1.3499999999999999e227 < t

   1. Initial program 82.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 81.8

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

   5. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

Alternative 13: 57.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{elif}\;t \leq 18:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= t 2.9e-261)
   (/
    2.0
    (*
     (/ (* (* (* k t) (* k t)) 2.0) (* (fma -0.5 (* k k) 1.0) (* l_m l_m)))
     t))
   (if (<= t 18.0)
     (/
      2.0
      (*
       (/
        (*
         (fma (fma -0.6666666666666666 (* t t) 1.0) (* k k) (* (* t t) 2.0))
         (* k k))
        (* 1.0 (* l_m l_m)))
       t))
     (/ (* l_m l_m) (* (pow (* k t) 2.0) t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (t <= 2.9e-261) {
		tmp = 2.0 / (((((k * t) * (k * t)) * 2.0) / (fma(-0.5, (k * k), 1.0) * (l_m * l_m))) * t);
	} else if (t <= 18.0) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / (1.0 * (l_m * l_m))) * t);
	} else {
		tmp = (l_m * l_m) / (pow((k * t), 2.0) * t);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (t <= 2.9e-261)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0) / Float64(fma(-0.5, Float64(k * k), 1.0) * Float64(l_m * l_m))) * t));
	elseif (t <= 18.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / Float64(1.0 * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t) ^ 2.0) * t));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[t, 2.9e-261], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 18.0], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.89999999999999985e-261

   1. Initial program 53.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 54.0

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 54.0%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 53.1

          \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 53.1%

       \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 53.1

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

   13. Applied rewrites 53.1%

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

   if 2.89999999999999985e-261 < t < 18

   1. Initial program 49.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 51.9

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 53.8

           \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 53.8%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

   14. Applied rewrites 71.5%

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

   if 18 < t

   1. Initial program 66.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 59.4

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

   5. Applied rewrites 59.4%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 67.5

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

   9. Applied rewrites 67.5%

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

Alternative 14: 54.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.8e+134)
   (/
    2.0
    (*
     (/ (* (* (* k t) (* k t)) 2.0) (* (fma -0.5 (* k k) 1.0) (* l_m l_m)))
     t))
   (/ (* l_m l_m) (* k (* k (pow t 3.0))))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.8e+134) {
		tmp = 2.0 / (((((k * t) * (k * t)) * 2.0) / (fma(-0.5, (k * k), 1.0) * (l_m * l_m))) * t);
	} else {
		tmp = (l_m * l_m) / (k * (k * pow(t, 3.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.8e+134)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0) / Float64(fma(-0.5, Float64(k * k), 1.0) * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64(k * Float64(k * (t ^ 3.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.8e+134], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.80000000000000011e134

   1. Initial program 56.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 59.1

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 59.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 59.4

          \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 59.4%

       \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 59.4

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

   13. Applied rewrites 59.4%

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

   if 4.80000000000000011e134 < k

   1. Initial program 51.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 51.5

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

   5. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 51.8

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

   7. Applied rewrites 51.8%

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

Alternative 15: 54.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(l\_m \cdot l\_m\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 4.8e+134)
   (/
    2.0
    (*
     (/ (* (* (* k t) (* k t)) 2.0) (* (fma -0.5 (* k k) 1.0) (* l_m l_m)))
     t))
   (/ (* l_m l_m) (* (* k k) (* (* t t) t)))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	double tmp;
	if (k <= 4.8e+134) {
		tmp = 2.0 / (((((k * t) * (k * t)) * 2.0) / (fma(-0.5, (k * k), 1.0) * (l_m * l_m))) * t);
	} else {
		tmp = (l_m * l_m) / ((k * k) * ((t * t) * t));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, l_m, k)
	tmp = 0.0
	if (k <= 4.8e+134)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0) / Float64(fma(-0.5, Float64(k * k), 1.0) * Float64(l_m * l_m))) * t));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t * t) * t)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := If[LessEqual[k, 4.8e+134], N[(2.0 / N[(N[(N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.80000000000000011e134

   1. Initial program 56.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 59.1

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 59.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 59.4

          \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 59.4%

       \[\leadsto \frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 59.4

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

   13. Applied rewrites 59.4%

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

   if 4.80000000000000011e134 < k

   1. Initial program 51.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 51.5

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

   5. Applied rewrites 51.5%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

Alternative 16: 51.4% accurate, 12.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (/ (* l_m l_m) (* (* k k) (* (* t t) t))))

C

l_m = fabs(l);
double code(double t, double l_m, double k) {
	return (l_m * l_m) / ((k * k) * ((t * t) * t));
}

Fortran

l_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_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = (l_m * l_m) / ((k * k) * ((t * t) * t))
end function

Java

l_m = Math.abs(l);
public static double code(double t, double l_m, double k) {
	return (l_m * l_m) / ((k * k) * ((t * t) * t));
}

Python

l_m = math.fabs(l)
def code(t, l_m, k):
	return (l_m * l_m) / ((k * k) * ((t * t) * t))

Julia

l_m = abs(l)
function code(t, l_m, k)
	return Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t * t) * t)))
end

MATLAB

l_m = abs(l);
function tmp = code(t, l_m, k)
	tmp = (l_m * l_m) / ((k * k) * ((t * t) * t));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, k_] := N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 53.8

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

5. Applied rewrites 53.8%

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

   1. lift-pow.f64 N/A

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

   2. pow3 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 53.8

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

7. Applied rewrites 53.8%

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

Reproduce

herbie shell --seed 2025064 
(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))))