Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 83.6%

Time: 8.1s

Alternatives: 19

Speedup: 7.8×

• 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: 54.7% 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: 83.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (/
   (*
    (/ t l)
    (/ (fma (pow (* (sin k) t) 2.0) 2.0 (pow (* (sin k) k) 2.0)) (cos k)))
   l)))

C

double code(double t, double l, double k) {
	return 2.0 / (((t / l) * (fma(pow((sin(k) * t), 2.0), 2.0, pow((sin(k) * k), 2.0)) / cos(k))) / l);
}

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(fma((Float64(sin(k) * t) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) / cos(k))) / l))
end

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 76.3%

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

7. Applied rewrites 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 81.7

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

9. Applied rewrites 81.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-fma.f64 N/A

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

    6. lift-pow.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-sin.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lift-cos.f64 N/A

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

    13. associate-*l/ N/A

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

11. Applied rewrites 84.2%

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

Alternative 2: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))
      5e+302)
   (/
    (/ 2.0 (* (/ (/ (* (* t t) t) l) l) (* (sin k) (tan k))))
    (fma (/ k t) (/ k t) 2.0))
   (/ 2.0 (* (/ (/ t l) l) (/ (* (pow (* k t) 2.0) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)) <= 5e+302) {
		tmp = (2.0 / (((((t * t) * t) / l) / l) * (sin(k) * tan(k)))) / fma((k / t), (k / t), 2.0);
	} else {
		tmp = 2.0 / (((t / l) / l) * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (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)) <= 5e+302)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * t) * t) / l) / l) * Float64(sin(k) * tan(k)))) / fma(Float64(k / t), Float64(k / t), 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], 5e+302], N[(N[(2.0 / N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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))) < 5e302

   1. Initial program 84.8%

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

   4. Applied rewrites 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-+l+ N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 82.1

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

   6. Applied rewrites 82.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if 5e302 < (*.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 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 63.1%

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

   7. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.3

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

   9. Applied rewrites 71.3%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 60.2

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

   12. Applied rewrites 60.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sin k \cdot t\right)}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \frac{\mathsf{fma}\left(t\_1, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{t\_1 \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (* (sin k) t) 2.0)))
   (if (<= (* l l) 5e+232)
     (/
      2.0
      (* (/ t (* l l)) (/ (fma t_1 2.0 (pow (* (sin k) k) 2.0)) (cos k))))
     (/ 2.0 (* (/ (/ t l) l) (/ (* t_1 2.0) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((sin(k) * t), 2.0);
	double tmp;
	if ((l * l) <= 5e+232) {
		tmp = 2.0 / ((t / (l * l)) * (fma(t_1, 2.0, pow((sin(k) * k), 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (((t / l) / l) * ((t_1 * 2.0) / cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * t) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 5e+232)
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(fma(t_1, 2.0, (Float64(sin(k) * k) ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64(t_1 * 2.0) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e+232], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$1 * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999987e232

   1. Initial program 68.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 86.5%

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

   7. Applied rewrites 87.6%

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

   if 4.99999999999999987e232 < (*.f64 l l)

   1. Initial program 41.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.7%

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

   7. Applied rewrites 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 65.5

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

   9. Applied rewrites 65.5%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 66.3

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

   12. Applied rewrites 66.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sin k \cdot t\right)}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{+220}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, t\_1, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{t\_1 \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (* (sin k) t) 2.0)))
   (if (<= (* l l) 1e+220)
     (/
      2.0
      (* (/ (fma 2.0 t_1 (pow (* (sin k) k) 2.0)) (* (cos k) (* l l))) t))
     (/ 2.0 (* (/ (/ t l) l) (/ (* t_1 2.0) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((sin(k) * t), 2.0);
	double tmp;
	if ((l * l) <= 1e+220) {
		tmp = 2.0 / ((fma(2.0, t_1, pow((sin(k) * k), 2.0)) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((t / l) / l) * ((t_1 * 2.0) / cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * t) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 1e+220)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, t_1, (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64(t_1 * 2.0) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e+220], N[(2.0 / N[(N[(N[(2.0 * t$95$1 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$1 * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e220

   1. Initial program 69.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 86.9%

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

   if 1e220 < (*.f64 l l)

   1. Initial program 41.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.7%

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

   7. Applied rewrites 56.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 66.4

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 66.0

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

   12. Applied rewrites 66.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+220}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 9.5e+100)
   (/
    2.0
    (/
     (* (fma (pow (* (sin k) t) 2.0) 2.0 (pow (* (sin k) k) 2.0)) t)
     (* (* (cos k) l) l)))
   (/ 2.0 (* (/ (/ t l) l) (/ (* (pow (* k t) 2.0) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e+100) {
		tmp = 2.0 / ((fma(pow((sin(k) * t), 2.0), 2.0, pow((sin(k) * k), 2.0)) * t) / ((cos(k) * l) * l));
	} else {
		tmp = 2.0 / (((t / l) / l) * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 9.5e+100)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k) * t) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0)) * t) / Float64(Float64(cos(k) * l) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 9.5e+100], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999995e100

   1. Initial program 60.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.7%

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

   7. Applied rewrites 79.4%

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

   if 9.4999999999999995e100 < t

   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 74.9%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 85.0

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

   9. Applied rewrites 85.0%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 81.6

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

   12. Applied rewrites 81.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (/
   (* (/ t l) (fma (pow (* (sin k) t) 2.0) 2.0 (pow (* (sin k) k) 2.0)))
   (* l (cos k)))))

C

double code(double t, double l, double k) {
	return 2.0 / (((t / l) * fma(pow((sin(k) * t), 2.0), 2.0, pow((sin(k) * k), 2.0))) / (l * cos(k)));
}

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(t / l) * fma((Float64(sin(k) * t) ^ 2.0), 2.0, (Float64(sin(k) * k) ^ 2.0))) / Float64(l * cos(k))))
end

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 76.3%

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

7. Applied rewrites 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 81.7

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

9. Applied rewrites 81.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-fma.f64 N/A

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

    6. lift-pow.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-sin.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-sin.f64 N/A

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

    12. lift-cos.f64 N/A

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

    13. frac-times N/A

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

11. Applied rewrites 84.2%

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

Alternative 7: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = (l * l) / ((k * k) * ((t * t) * t));
	} else {
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0)) <= math.inf:
		tmp = (l * l) / ((k * k) * ((t * t) * t))
	else:
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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)) <= Inf)
		tmp = Float64(Float64(l * l) / Float64(Float64(k * k) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * Float64(k * k)) / Float64(l * l)) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0)) <= Inf)
		tmp = (l * l) / ((k * k) * ((t * t) * t));
	else
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], Infinity], N[(N[(l * l), $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[(l * l), $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 83.1%

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

   3. Taylor expanded in 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 76.2

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

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

   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 38.2%

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

   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. unpow2 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. lower-*.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. unpow2 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. lower-*.f64 N/A

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

      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 28.1

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

   8. Applied rewrites 28.1%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 28.0

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

   11. Applied rewrites 28.0%

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

   12. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 29.9

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

   14. Applied rewrites 29.9%

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

Alternative 8: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 1.15e-60)
     (/ 2.0 (* t_1 (/ (pow (* (sin k) k) 2.0) (cos k))))
     (if (<= t 2.15e+80)
       (/
        2.0
        (*
         (* (/ (/ (pow t 3.0) l) l) (sin k))
         (* (tan k) (+ (+ (pow (/ k t) 2.0) 1.0) 1.0))))
       (/ 2.0 (* t_1 (/ (* (pow (* (sin k) t) 2.0) 2.0) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 1.15e-60) {
		tmp = 2.0 / (t_1 * (pow((sin(k) * k), 2.0) / cos(k)));
	} else if (t <= 2.15e+80) {
		tmp = 2.0 / ((((pow(t, 3.0) / l) / l) * sin(k)) * (tan(k) * ((pow((k / t), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / (t_1 * ((pow((sin(k) * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) / l
    if (t <= 1.15d-60) then
        tmp = 2.0d0 / (t_1 * (((sin(k) * k) ** 2.0d0) / cos(k)))
    else if (t <= 2.15d+80) then
        tmp = 2.0d0 / (((((t ** 3.0d0) / l) / l) * sin(k)) * (tan(k) * ((((k / t) ** 2.0d0) + 1.0d0) + 1.0d0)))
    else
        tmp = 2.0d0 / (t_1 * ((((sin(k) * t) ** 2.0d0) * 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 1.15e-60) {
		tmp = 2.0 / (t_1 * (Math.pow((Math.sin(k) * k), 2.0) / Math.cos(k)));
	} else if (t <= 2.15e+80) {
		tmp = 2.0 / ((((Math.pow(t, 3.0) / l) / l) * Math.sin(k)) * (Math.tan(k) * ((Math.pow((k / t), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / (t_1 * ((Math.pow((Math.sin(k) * t), 2.0) * 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (t / l) / l
	tmp = 0
	if t <= 1.15e-60:
		tmp = 2.0 / (t_1 * (math.pow((math.sin(k) * k), 2.0) / math.cos(k)))
	elif t <= 2.15e+80:
		tmp = 2.0 / ((((math.pow(t, 3.0) / l) / l) * math.sin(k)) * (math.tan(k) * ((math.pow((k / t), 2.0) + 1.0) + 1.0)))
	else:
		tmp = 2.0 / (t_1 * ((math.pow((math.sin(k) * t), 2.0) * 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 1.15e-60)
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))));
	elseif (t <= 2.15e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / l) / l) * sin(k)) * Float64(tan(k) * Float64(Float64((Float64(k / t) ^ 2.0) + 1.0) + 1.0))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64((Float64(sin(k) * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (t / l) / l;
	tmp = 0.0;
	if (t <= 1.15e-60)
		tmp = 2.0 / (t_1 * (((sin(k) * k) ^ 2.0) / cos(k)));
	elseif (t <= 2.15e+80)
		tmp = 2.0 / (((((t ^ 3.0) / l) / l) * sin(k)) * (tan(k) * ((((k / t) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = 2.0 / (t_1 * ((((sin(k) * t) ^ 2.0) * 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 1.15e-60], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e+80], N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1500000000000001e-60

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.4%

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

   7. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 81.2

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

   9. Applied rewrites 81.2%

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

   10. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 67.4

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

   12. Applied rewrites 67.4%

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

   if 1.1500000000000001e-60 < t < 2.15000000000000002e80

   1. Initial program 78.0%

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

   4. Applied rewrites 78.3%

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

   if 2.15000000000000002e80 < t

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.9%

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

   7. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.2

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

   9. Applied rewrites 86.2%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 83.2

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

   12. Applied rewrites 83.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 1.8e-35)
     (/ 2.0 (* t_1 (/ (pow (* (sin k) k) 2.0) (cos k))))
     (if (<= t 8e+75)
       (/
        2.0
        (*
         (* (* (/ (* (* t t) t) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
       (/ 2.0 (* t_1 (/ (* (pow (* (sin k) t) 2.0) 2.0) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 1.8e-35) {
		tmp = 2.0 / (t_1 * (pow((sin(k) * k), 2.0) / cos(k)));
	} else if (t <= 8e+75) {
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (t_1 * ((pow((sin(k) * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) / l
    if (t <= 1.8d-35) then
        tmp = 2.0d0 / (t_1 * (((sin(k) * k) ** 2.0d0) / cos(k)))
    else if (t <= 8d+75) then
        tmp = 2.0d0 / ((((((t * t) * t) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (t_1 * ((((sin(k) * t) ** 2.0d0) * 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 1.8e-35) {
		tmp = 2.0 / (t_1 * (Math.pow((Math.sin(k) * k), 2.0) / Math.cos(k)));
	} else if (t <= 8e+75) {
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (t_1 * ((Math.pow((Math.sin(k) * t), 2.0) * 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (t / l) / l
	tmp = 0
	if t <= 1.8e-35:
		tmp = 2.0 / (t_1 * (math.pow((math.sin(k) * k), 2.0) / math.cos(k)))
	elif t <= 8e+75:
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (t_1 * ((math.pow((math.sin(k) * t), 2.0) * 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 1.8e-35)
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))));
	elseif (t <= 8e+75)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64((Float64(sin(k) * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (t / l) / l;
	tmp = 0.0;
	if (t <= 1.8e-35)
		tmp = 2.0 / (t_1 * (((sin(k) * k) ^ 2.0) / cos(k)));
	elseif (t <= 8e+75)
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (t_1 * ((((sin(k) * t) ^ 2.0) * 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 1.8e-35], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e+75], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $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], N[(2.0 / N[(t$95$1 * N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.80000000000000009e-35

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.3%

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

   7. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 81.0

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

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 67.5

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

   12. Applied rewrites 67.5%

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

   if 1.80000000000000009e-35 < t < 7.99999999999999941e75

   1. Initial program 79.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

   if 7.99999999999999941e75 < t

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.9%

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

   7. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.2

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

   9. Applied rewrites 86.2%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 83.2

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

   12. Applied rewrites 83.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 4e-22)
     (/ 2.0 (* t_1 (/ (pow (* (sin k) k) 2.0) (cos k))))
     (/ 2.0 (* t_1 (/ (* (pow (* k t) 2.0) 2.0) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 4e-22) {
		tmp = 2.0 / (t_1 * (pow((sin(k) * k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / (t_1 * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) / l
    if (t <= 4d-22) then
        tmp = 2.0d0 / (t_1 * (((sin(k) * k) ** 2.0d0) / cos(k)))
    else
        tmp = 2.0d0 / (t_1 * ((((k * t) ** 2.0d0) * 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 4e-22) {
		tmp = 2.0 / (t_1 * (Math.pow((Math.sin(k) * k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / (t_1 * ((Math.pow((k * t), 2.0) * 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (t / l) / l
	tmp = 0
	if t <= 4e-22:
		tmp = 2.0 / (t_1 * (math.pow((math.sin(k) * k), 2.0) / math.cos(k)))
	else:
		tmp = 2.0 / (t_1 * ((math.pow((k * t), 2.0) * 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 4e-22)
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (t / l) / l;
	tmp = 0.0;
	if (t <= 4e-22)
		tmp = 2.0 / (t_1 * (((sin(k) * k) ^ 2.0) / cos(k)));
	else
		tmp = 2.0 / (t_1 * ((((k * t) ^ 2.0) * 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 4e-22], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000002e-22

   1. Initial program 58.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.0%

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

   7. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 81.2

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

   9. Applied rewrites 81.2%

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

   10. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 67.3

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

   12. Applied rewrites 67.3%

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

   if 4.0000000000000002e-22 < t

   1. Initial program 65.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 77.0%

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

   7. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.9

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

   9. Applied rewrites 82.9%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 75.3

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

   12. Applied rewrites 75.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 4e-22)
   (/ 2.0 (* (/ t (* l l)) (/ (pow (* (sin k) k) 2.0) (cos k))))
   (/ 2.0 (* (/ (/ t l) l) (/ (* (pow (* k t) 2.0) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 4e-22) {
		tmp = 2.0 / ((t / (l * l)) * (pow((sin(k) * k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / (((t / l) / l) * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 4d-22) then
        tmp = 2.0d0 / ((t / (l * l)) * (((sin(k) * k) ** 2.0d0) / cos(k)))
    else
        tmp = 2.0d0 / (((t / l) / l) * ((((k * t) ** 2.0d0) * 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 4e-22) {
		tmp = 2.0 / ((t / (l * l)) * (Math.pow((Math.sin(k) * k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((t / l) / l) * ((Math.pow((k * t), 2.0) * 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 4e-22:
		tmp = 2.0 / ((t / (l * l)) * (math.pow((math.sin(k) * k), 2.0) / math.cos(k)))
	else:
		tmp = 2.0 / (((t / l) / l) * ((math.pow((k * t), 2.0) * 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 4e-22)
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64((Float64(sin(k) * k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 4e-22)
		tmp = 2.0 / ((t / (l * l)) * (((sin(k) * k) ^ 2.0) / cos(k)));
	else
		tmp = 2.0 / (((t / l) / l) * ((((k * t) ^ 2.0) * 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 4e-22], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000002e-22

   1. Initial program 58.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.0%

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

   7. Applied rewrites 78.4%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 66.1

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

   10. Applied rewrites 66.1%

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

   if 4.0000000000000002e-22 < t

   1. Initial program 65.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 77.0%

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

   7. Applied rewrites 76.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.9

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

   9. Applied rewrites 82.9%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 75.3

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

   12. Applied rewrites 75.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.8e-35)
   (* (* (/ (* l l) (* k k)) (/ (cos k) (* (pow (sin k) 2.0) t))) 2.0)
   (/ 2.0 (* (/ (/ t l) l) (/ (* (pow (* k t) 2.0) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.8e-35) {
		tmp = (((l * l) / (k * k)) * (cos(k) / (pow(sin(k), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.8d-35) then
        tmp = (((l * l) / (k * k)) * (cos(k) / ((sin(k) ** 2.0d0) * t))) * 2.0d0
    else
        tmp = 2.0d0 / (((t / l) / l) * ((((k * t) ** 2.0d0) * 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.8e-35) {
		tmp = (((l * l) / (k * k)) * (Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * ((Math.pow((k * t), 2.0) * 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.8e-35:
		tmp = (((l * l) / (k * k)) * (math.cos(k) / (math.pow(math.sin(k), 2.0) * t))) * 2.0
	else:
		tmp = 2.0 / (((t / l) / l) * ((math.pow((k * t), 2.0) * 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.8e-35)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.8e-35)
		tmp = (((l * l) / (k * k)) * (cos(k) / ((sin(k) ^ 2.0) * t))) * 2.0;
	else
		tmp = 2.0 / (((t / l) / l) * ((((k * t) ^ 2.0) * 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.8e-35], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8e-35

   1. Initial program 57.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.7%

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

   if 2.8e-35 < t

   1. Initial program 66.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.4%

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

   7. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 83.3

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

   9. Applied rewrites 83.3%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 74.8

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

   12. Applied rewrites 74.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.8e-35)
   (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (cos k) (* l l))) t))
   (/ 2.0 (* (/ (/ t l) l) (/ (* (pow (* k t) 2.0) 2.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.8e-35) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((t / l) / l) * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.8d-35) then
        tmp = 2.0d0 / ((((sin(k) * k) ** 2.0d0) / (cos(k) * (l * l))) * t)
    else
        tmp = 2.0d0 / (((t / l) / l) * ((((k * t) ** 2.0d0) * 2.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.8e-35) {
		tmp = 2.0 / ((Math.pow((Math.sin(k) * k), 2.0) / (Math.cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((t / l) / l) * ((Math.pow((k * t), 2.0) * 2.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.8e-35:
		tmp = 2.0 / ((math.pow((math.sin(k) * k), 2.0) / (math.cos(k) * (l * l))) * t)
	else:
		tmp = 2.0 / (((t / l) / l) * ((math.pow((k * t), 2.0) * 2.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.8e-35)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.8e-35)
		tmp = 2.0 / ((((sin(k) * k) ^ 2.0) / (cos(k) * (l * l))) * t);
	else
		tmp = 2.0 / (((t / l) / l) * ((((k * t) ^ 2.0) * 2.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.8e-35], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8e-35

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.3%

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

   6. Taylor expanded in 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 64.9

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

   8. Applied rewrites 64.9%

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

   if 2.8e-35 < t

   1. Initial program 66.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.4%

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

   7. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 83.3

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

   9. Applied rewrites 83.3%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 74.8

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

   12. Applied rewrites 74.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 1.2e+14)
     (/
      2.0
      (*
       t_1
       (*
        (fma
         (+ (fma (* t t) -0.6666666666666666 1.0) (* t t))
         (* k k)
         (* (* t t) 2.0))
        (* k k))))
     (/ 2.0 (* t_1 (/ (* (pow (* k t) 2.0) 2.0) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 1.2e+14) {
		tmp = 2.0 / (t_1 * (fma((fma((t * t), -0.6666666666666666, 1.0) + (t * t)), (k * k), ((t * t) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (t_1 * ((pow((k * t), 2.0) * 2.0) / cos(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 1.2e+14)
		tmp = Float64(2.0 / Float64(t_1 * Float64(fma(Float64(fma(Float64(t * t), -0.6666666666666666, 1.0) + Float64(t * t)), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / cos(k))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 1.2e+14], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e14

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 77.0%

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

   7. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.0

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

   9. Applied rewrites 82.0%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 57.8%

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

   if 1.2e14 < t

   1. Initial program 61.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.3%

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

   7. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 81.0

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

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 72.9

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

   12. Applied rewrites 72.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left({\left(k \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 9.5e+14)
     (/
      2.0
      (*
       t_1
       (*
        (fma
         (+ (fma (* t t) -0.6666666666666666 1.0) (* t t))
         (* k k)
         (* (* t t) 2.0))
        (* k k))))
     (/ 2.0 (* t_1 (* (pow (* k t) 2.0) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 9.5e+14) {
		tmp = 2.0 / (t_1 * (fma((fma((t * t), -0.6666666666666666, 1.0) + (t * t)), (k * k), ((t * t) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / (t_1 * (pow((k * t), 2.0) * 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 9.5e+14)
		tmp = Float64(2.0 / Float64(t_1 * Float64(fma(Float64(fma(Float64(t * t), -0.6666666666666666, 1.0) + Float64(t * t)), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(k * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 9.5e+14], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5e14

   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 77.1%

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

   7. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.1

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

   9. Applied rewrites 82.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 57.5%

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

   if 9.5e14 < t

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.0%

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 67.6

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

   10. Applied rewrites 67.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 73.7

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

   12. Applied rewrites 73.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 9.2e+14)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (+ (fma (* t t) -0.6666666666666666 1.0) (* t t))
       (* k k)
       (* (* t t) 2.0))
      (* k k))))
   (/ 2.0 (* (/ t (* l l)) (* (* (* k t) (* k t)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.2e+14) {
		tmp = 2.0 / (((t / l) / l) * (fma((fma((t * t), -0.6666666666666666, 1.0) + (t * t)), (k * k), ((t * t) * 2.0)) * (k * k)));
	} else {
		tmp = 2.0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 9.2e+14)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(Float64(fma(Float64(t * t), -0.6666666666666666, 1.0) + Float64(t * t)), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 9.2e+14], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.2e14

   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 77.1%

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

   7. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.1

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

   9. Applied rewrites 82.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 57.5%

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

   if 9.2e14 < t

   1. Initial program 61.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.0%

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 67.6

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

   10. Applied rewrites 67.6%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 67.6

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

   12. Applied rewrites 67.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right) \cdot k\right) \cdot k}{\ell \cdot \ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 4.5e+98)
   (/
    2.0
    (*
     (/
      (*
       (*
        (fma (* (fma (* t t) -0.6666666666666666 1.0) k) k (* (* t t) 2.0))
        k)
       k)
      (* l l))
     t))
   (/ 2.0 (* (/ t (* l l)) (* (* (* k t) (* k t)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 4.5e+98) {
		tmp = 2.0 / ((((fma((fma((t * t), -0.6666666666666666, 1.0) * k), k, ((t * t) * 2.0)) * k) * k) / (l * l)) * t);
	} else {
		tmp = 2.0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 4.5e+98)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(t * t), -0.6666666666666666, 1.0) * k), k, Float64(Float64(t * t) * 2.0)) * k) * k) / Float64(l * l)) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 4.5e+98], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5000000000000002e98

   1. Initial program 60.3%

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

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   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. unpow2 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. lower-*.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. unpow2 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. lower-*.f64 N/A

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

      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 55.1

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 54.9

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

   11. Applied rewrites 54.9%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

   13. Applied rewrites 56.9%

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

   if 4.5000000000000002e98 < t

   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 74.9%

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 74.0

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

   10. Applied rewrites 74.0%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 74.0

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

   12. Applied rewrites 74.0%

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

Alternative 18: 57.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.3e-60)
   (/ 2.0 (* (/ (* (* k k) (* k k)) (* l l)) t))
   (/ 2.0 (* (/ t (* l l)) (* (* (* k t) (* k t)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.3e-60) {
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t);
	} else {
		tmp = 2.0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.3d-60) then
        tmp = 2.0d0 / ((((k * k) * (k * k)) / (l * l)) * t)
    else
        tmp = 2.0d0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.3e-60) {
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t);
	} else {
		tmp = 2.0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.3e-60:
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t)
	else:
		tmp = 2.0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.3e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * Float64(k * k)) / Float64(l * l)) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(Float64(k * t) * Float64(k * t)) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.3e-60)
		tmp = 2.0 / ((((k * k) * (k * k)) / (l * l)) * t);
	else
		tmp = 2.0 / ((t / (l * l)) * (((k * t) * (k * t)) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.3e-60], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2999999999999999e-60

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.4%

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

   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. unpow2 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. lower-*.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. unpow2 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. lower-*.f64 N/A

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

      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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 54.2

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

   11. Applied rewrites 54.2%

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

   12. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 59.7

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

   14. Applied rewrites 59.7%

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

   if 1.2999999999999999e-60 < t

   1. Initial program 66.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.2%

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 69.3

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

   10. Applied rewrites 69.3%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 69.3

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

   12. Applied rewrites 69.3%

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

Alternative 19: 51.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* l l) (* (* k k) (* (* t t) t))))

C

double code(double t, double l, double k) {
	return (l * l) / ((k * k) * ((t * t) * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) / ((k * k) * ((t * t) * t))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) / ((k * k) * ((t * t) * t));
}

Python

def code(t, l, k):
	return (l * l) / ((k * k) * ((t * t) * t))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) / Float64(Float64(k * k) * Float64(Float64(t * t) * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) / ((k * k) * ((t * t) * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 58.6

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

5. Applied rewrites 58.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
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. unpow3 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 58.6

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

7. Applied rewrites 58.6%

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

Reproduce

herbie shell --seed 2025060 
(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))))