Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 88.7%

Time: 7.7s

Alternatives: 25

Speedup: 3.4×

• 27.2% 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: 55.2% 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: 88.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-neg N/A

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

   5. metadata-eval N/A

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

   6. pow-flip N/A

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

   7. lower-/.f64 N/A

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

   8. pow-flip N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 73.8

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

7. Applied rewrites 73.8%

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

9. Applied rewrites 79.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-sin.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-cos.f64 N/A

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

    9. associate-/r* N/A

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

    10. lower-/.f64 N/A

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

11. Applied rewrites 88.7%

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

Alternative 2: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))))
   (if (<= t_1 -5e-250)
     (/ 2.0 (* (/ (* (* (* t t) 2.0) (* k k)) (* (cos k) (* l l))) t))
     (if (<= t_1 2e+303)
       (/ 2.0 (* (* (/ (pow (* k t) 2.0) (* l l)) 2.0) t))
       (/ 2.0 (* (pow (/ k l) 2.0) (* (* k k) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = 2.0 / (((((t * t) * 2.0) * (k * k)) / (cos(k) * (l * l))) * t);
	} else if (t_1 <= 2e+303) {
		tmp = 2.0 / (((pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (pow((k / l), 2.0) * ((k * k) * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    if (t_1 <= (-5d-250)) then
        tmp = 2.0d0 / (((((t * t) * 2.0d0) * (k * k)) / (cos(k) * (l * l))) * t)
    else if (t_1 <= 2d+303) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) / (l * l)) * 2.0d0) * t)
    else
        tmp = 2.0d0 / (((k / l) ** 2.0d0) * ((k * k) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 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));
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = 2.0 / (((((t * t) * 2.0) * (k * k)) / (Math.cos(k) * (l * l))) * t);
	} else if (t_1 <= 2e+303) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (Math.pow((k / l), 2.0) * ((k * k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 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))
	tmp = 0
	if t_1 <= -5e-250:
		tmp = 2.0 / (((((t * t) * 2.0) * (k * k)) / (math.cos(k) * (l * l))) * t)
	elif t_1 <= 2e+303:
		tmp = 2.0 / (((math.pow((k * t), 2.0) / (l * l)) * 2.0) * t)
	else:
		tmp = 2.0 / (math.pow((k / l), 2.0) * ((k * k) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = 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)))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * t) * 2.0) * Float64(k * k)) / Float64(cos(k) * Float64(l * l))) * t));
	elseif (t_1 <= 2e+303)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64(Float64(k * k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	tmp = 0.0;
	if (t_1 <= -5e-250)
		tmp = 2.0 / (((((t * t) * 2.0) * (k * k)) / (cos(k) * (l * l))) * t);
	elseif (t_1 <= 2e+303)
		tmp = 2.0 / (((((k * t) ^ 2.0) / (l * l)) * 2.0) * t);
	else
		tmp = 2.0 / (((k / l) ^ 2.0) * ((k * k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e-250], N[(2.0 / N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -5.00000000000000027e-250

   1. Initial program 89.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 77.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 76.1

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

   11. Applied rewrites 76.1%

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

   if -5.00000000000000027e-250 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 79.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 83.1

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

   8. Applied rewrites 83.1%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 21.8%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 45.5

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

   8. Applied rewrites 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 52.3

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

   10. Applied rewrites 52.3%

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

Alternative 3: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))))
   (if (<= t_1 -5e-250)
     (/
      2.0
      (*
       (/
        (*
         (fma
          (+
           (fma
            (- (* (* t t) 0.08888888888888889) 0.3333333333333333)
            (* k k)
            (* -0.6666666666666666 (* t t)))
           1.0)
          (* k k)
          (* (* t t) 2.0))
         (* k k))
        (* (fma (- (* 0.041666666666666664 (* k k)) 0.5) (* k k) 1.0) (* l l)))
       t))
     (if (<= t_1 2e+303)
       (/ 2.0 (* (* (/ (pow (* k t) 2.0) (* l l)) 2.0) t))
       (/ 2.0 (* (pow (/ k l) 2.0) (* (* k k) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = 2.0 / (((fma((fma((((t * t) * 0.08888888888888889) - 0.3333333333333333), (k * k), (-0.6666666666666666 * (t * t))) + 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / (fma(((0.041666666666666664 * (k * k)) - 0.5), (k * k), 1.0) * (l * l))) * t);
	} else if (t_1 <= 2e+303) {
		tmp = 2.0 / (((pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (pow((k / l), 2.0) * ((k * k) * t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = 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)))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(Float64(Float64(Float64(t * t) * 0.08888888888888889) - 0.3333333333333333), Float64(k * k), Float64(-0.6666666666666666 * Float64(t * t))) + 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / Float64(fma(Float64(Float64(0.041666666666666664 * Float64(k * k)) - 0.5), Float64(k * k), 1.0) * Float64(l * l))) * t));
	elseif (t_1 <= 2e+303)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64(Float64(k * k) * t)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e-250], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 0.08888888888888889), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(-0.6666666666666666 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.041666666666666664 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -5.00000000000000027e-250

   1. Initial program 89.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 77.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(1 + {k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right)\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left({k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) \cdot {k}^{2} + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 77.0

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 77.0%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4}{45} \cdot {t}^{2} - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left({t}^{2} \cdot \frac{4}{45} - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left({t}^{2} \cdot \frac{4}{45} - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{4}{45} - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lift-*.f64 76.6

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 76.6%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if -5.00000000000000027e-250 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 79.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 83.1

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

   8. Applied rewrites 83.1%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 21.8%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 45.5

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

   8. Applied rewrites 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 52.3

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

   10. Applied rewrites 52.3%

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

Alternative 4: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+303)
   (/
    2.0
    (*
     (/
      (fma 2.0 (pow (* (sin k) t) 2.0) (pow (* (sin k) k) 2.0))
      (* (cos k) (* l l)))
     t))
   (/ 2.0 (/ (* (pow (/ k l) 2.0) (* (pow (sin k) 2.0) t)) (cos k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))) <= 2e+303) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / ((pow((k / l), 2.0) * (pow(sin(k), 2.0) * t)) / cos(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= 2e+303)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k / l) ^ 2.0) * Float64((sin(k) ^ 2.0) * t)) / cos(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], 2e+303], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 81.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.3%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 21.8%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 77.8%

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

Alternative 5: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{\ell}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(t\_2 \cdot \frac{t}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_1, \frac{t\_2}{\cos k}, \frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2}{\left(\cos k \cdot \ell\right) \cdot \ell}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k l) 2.0)) (t_2 (pow (sin k) 2.0)))
   (if (<= t 7.6e-70)
     (/ 2.0 (* t_1 (* t_2 (/ t (cos k)))))
     (if (<= t 3.6e+208)
       (/
        2.0
        (*
         (fma
          t_1
          (/ t_2 (cos k))
          (/ (* (pow (* (sin k) t) 2.0) 2.0) (* (* (cos k) l) l)))
         t))
       (/
        2.0
        (*
         (* (* (exp (- (* (log t) 3.0) (* (log l) 2.0))) (sin k)) (tan k))
         2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / l), 2.0);
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t <= 7.6e-70) {
		tmp = 2.0 / (t_1 * (t_2 * (t / cos(k))));
	} else if (t <= 3.6e+208) {
		tmp = 2.0 / (fma(t_1, (t_2 / cos(k)), ((pow((sin(k) * t), 2.0) * 2.0) / ((cos(k) * l) * l))) * t);
	} else {
		tmp = 2.0 / (((exp(((log(t) * 3.0) - (log(l) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / l) ^ 2.0
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= 7.6e-70)
		tmp = Float64(2.0 / Float64(t_1 * Float64(t_2 * Float64(t / cos(k)))));
	elseif (t <= 3.6e+208)
		tmp = Float64(2.0 / Float64(fma(t_1, Float64(t_2 / cos(k)), Float64(Float64((Float64(sin(k) * t) ^ 2.0) * 2.0) / Float64(Float64(cos(k) * l) * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t) * 3.0) - Float64(log(l) * 2.0))) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 7.6e-70], N[(2.0 / N[(t$95$1 * N[(t$95$2 * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e+208], N[(2.0 / N[(N[(t$95$1 * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.5999999999999995e-70

   1. Initial program 49.3%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 63.0

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

   5. Applied rewrites 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 75.1

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cos.f64 75.1

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

   7. Applied rewrites 75.1%

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

   if 7.5999999999999995e-70 < t < 3.60000000000000003e208

   1. Initial program 67.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 75.1%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. metadata-eval N/A

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

      6. pow-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 75.0

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

   7. Applied rewrites 75.0%

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

   9. Applied rewrites 84.3%

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

   if 3.60000000000000003e208 < t

   1. Initial program 69.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 43.6

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

   7. Applied rewrites 43.6%

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

Alternative 6: 68.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+303)
   (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
   (/ 2.0 (* (pow (/ k l) 2.0) (* (* k k) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))) <= 2e+303) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (pow((k / l), 2.0) * ((k * k) * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))) <= 2d+303) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else
        tmp = 2.0d0 / (((k / l) ** 2.0d0) * ((k * k) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= 2e+303) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (Math.pow((k / l), 2.0) * ((k * k) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= 2e+303:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	else:
		tmp = 2.0 / (math.pow((k / l), 2.0) * ((k * k) * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= 2e+303)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64(Float64(k * k) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0))) <= 2e+303)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	else
		tmp = 2.0 / (((k / l) ^ 2.0) * ((k * k) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], 2e+303], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 81.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 81.7

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

   8. Applied rewrites 81.7%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 21.8%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 45.5

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

   8. Applied rewrites 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 52.3

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

   10. Applied rewrites 52.3%

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

Alternative 7: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.5e+15)
   (/ 2.0 (/ (* (pow (/ k l) 2.0) (* (pow (sin k) 2.0) t)) (cos k)))
   (/
    2.0
    (*
     (* (* (exp (- (* (log t) 3.0) (* (log l) 2.0))) (sin k)) (tan k))
     (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.5e+15) {
		tmp = 2.0 / ((pow((k / l), 2.0) * (pow(sin(k), 2.0) * t)) / cos(k));
	} else {
		tmp = 2.0 / (((exp(((log(t) * 3.0) - (log(l) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
	}
	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 <= 5.5d+15) then
        tmp = 2.0d0 / ((((k / l) ** 2.0d0) * ((sin(k) ** 2.0d0) * t)) / cos(k))
    else
        tmp = 2.0d0 / (((exp(((log(t) * 3.0d0) - (log(l) * 2.0d0))) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.5e+15) {
		tmp = 2.0 / ((Math.pow((k / l), 2.0) * (Math.pow(Math.sin(k), 2.0) * t)) / Math.cos(k));
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t) * 3.0) - (Math.log(l) * 2.0))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.5e+15:
		tmp = 2.0 / ((math.pow((k / l), 2.0) * (math.pow(math.sin(k), 2.0) * t)) / math.cos(k))
	else:
		tmp = 2.0 / (((math.exp(((math.log(t) * 3.0) - (math.log(l) * 2.0))) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.5e+15)
		tmp = Float64(2.0 / Float64(Float64((Float64(k / l) ^ 2.0) * Float64((sin(k) ^ 2.0) * t)) / cos(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t) * 3.0) - Float64(log(l) * 2.0))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.5e+15)
		tmp = 2.0 / ((((k / l) ^ 2.0) * ((sin(k) ^ 2.0) * t)) / cos(k));
	else
		tmp = 2.0 / (((exp(((log(t) * 3.0) - (log(l) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.5e+15], N[(2.0 / N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision]), $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]]

Derivation

1. Split input into 2 regimes

2. if t < 5.5e15

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 63.9

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

   5. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 75.9%

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

   if 5.5e15 < t

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 44.3

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

   4. Applied rewrites 44.3%

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

Alternative 8: 58.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+303)
   (/ (* l l) (* (* k k) (* (* t t) t)))
   (/ 2.0 (* (/ (* k k) (* l l)) (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))) <= 2e+303) {
		tmp = (l * l) / ((k * k) * ((t * t) * t));
	} else {
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))) <= 2d+303) then
        tmp = (l * l) / ((k * k) * ((t * t) * t))
    else
        tmp = 2.0d0 / (((k * k) / (l * l)) * (k * (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((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))) <= 2e+303) {
		tmp = (l * l) / ((k * k) * ((t * t) * t));
	} else {
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (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))) <= 2e+303:
		tmp = (l * l) / ((k * k) * ((t * t) * t))
	else:
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (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))) <= 2e+303)
		tmp = Float64(Float64(l * l) / Float64(Float64(k * k) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0))) <= 2e+303)
		tmp = (l * l) / ((k * k) * ((t * t) * t));
	else
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[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], 2e+303], 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[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 81.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 68.5

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

   5. Applied rewrites 68.5%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 21.8%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 45.5

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

   8. Applied rewrites 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 45.5

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

   10. Applied rewrites 45.5%

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

Alternative 9: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 26500:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {k}^{4}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t\_1}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t\_1 \cdot t\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 26500.0)
     (/
      2.0
      (*
       (/ (fma 2.0 (pow (* (sin k) t) 2.0) (pow k 4.0)) (* (cos k) (* l l)))
       t))
     (if (<= k 9.2e+157)
       (/ 2.0 (/ (/ (* (* (* k k) t) t_1) (* (cos k) l)) l))
       (/ 2.0 (/ (* (pow (/ k l) 2.0) (* t_1 t)) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 26500.0) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t), 2.0), pow(k, 4.0)) / (cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((pow((k / l), 2.0) * (t_1 * t)) / cos(k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 26500.0)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t) ^ 2.0), (k ^ 4.0)) / Float64(cos(k) * Float64(l * l))) * t));
	elseif (k <= 9.2e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * t_1) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k / l) ^ 2.0) * Float64(t_1 * t)) / cos(k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 26500.0], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.2e+157], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 26500

   1. Initial program 57.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. lower-pow.f64 70.4

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {k}^{4}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 70.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {k}^{4}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 26500 < k < 9.20000000000000015e157

   1. Initial program 48.3%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 84.2%

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

   if 9.20000000000000015e157 < k

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 92.0%

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

Alternative 10: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 12500:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t\_1}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t\_1 \cdot t\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 12500.0)
     (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
     (if (<= k 9.2e+157)
       (/ 2.0 (/ (/ (* (* (* k k) t) t_1) (* (cos k) l)) l))
       (/ 2.0 (/ (* (pow (/ k l) 2.0) (* t_1 t)) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((pow((k / l), 2.0) * (t_1 * t)) / 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 = sin(k) ** 2.0d0
    if (k <= 12500.0d0) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else if (k <= 9.2d+157) then
        tmp = 2.0d0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l)
    else
        tmp = 2.0d0 / ((((k / l) ** 2.0d0) * (t_1 * t)) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (Math.cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((Math.pow((k / l), 2.0) * (t_1 * t)) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 12500.0:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	elif k <= 9.2e+157:
		tmp = 2.0 / (((((k * k) * t) * t_1) / (math.cos(k) * l)) / l)
	else:
		tmp = 2.0 / ((math.pow((k / l), 2.0) * (t_1 * t)) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 12500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	elseif (k <= 9.2e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * t_1) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k / l) ^ 2.0) * Float64(t_1 * t)) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 12500.0)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	elseif (k <= 9.2e+157)
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	else
		tmp = 2.0 / ((((k / l) ^ 2.0) * (t_1 * t)) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 12500.0], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.2e+157], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 12500

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 12500 < k < 9.20000000000000015e157

   1. Initial program 48.2%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   if 9.20000000000000015e157 < k

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 92.0%

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

Alternative 11: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 12500:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t\_1}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t\_1 \cdot \frac{t}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 12500.0)
     (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
     (if (<= k 9.2e+157)
       (/ 2.0 (/ (/ (* (* (* k k) t) t_1) (* (cos k) l)) l))
       (/ 2.0 (* (pow (/ k l) 2.0) (* t_1 (/ t (cos k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / (pow((k / l), 2.0) * (t_1 * (t / 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 = sin(k) ** 2.0d0
    if (k <= 12500.0d0) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else if (k <= 9.2d+157) then
        tmp = 2.0d0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l)
    else
        tmp = 2.0d0 / (((k / l) ** 2.0d0) * (t_1 * (t / cos(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (Math.cos(k) * l)) / l);
	} else {
		tmp = 2.0 / (Math.pow((k / l), 2.0) * (t_1 * (t / Math.cos(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 12500.0:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	elif k <= 9.2e+157:
		tmp = 2.0 / (((((k * k) * t) * t_1) / (math.cos(k) * l)) / l)
	else:
		tmp = 2.0 / (math.pow((k / l), 2.0) * (t_1 * (t / math.cos(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 12500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	elseif (k <= 9.2e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * t_1) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64(t_1 * Float64(t / cos(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 12500.0)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	elseif (k <= 9.2e+157)
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	else
		tmp = 2.0 / (((k / l) ^ 2.0) * (t_1 * (t / cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 12500.0], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.2e+157], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$1 * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 12500

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 12500 < k < 9.20000000000000015e157

   1. Initial program 48.2%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   if 9.20000000000000015e157 < k

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 92.1

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cos.f64 92.1

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

   7. Applied rewrites 92.1%

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

Alternative 12: 73.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 12500:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t\_1}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t\_1 \cdot t}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 12500.0)
     (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
     (if (<= k 9.2e+157)
       (/ 2.0 (/ (/ (* (* (* k k) t) t_1) (* (cos k) l)) l))
       (/ 2.0 (* (* (/ k l) (/ k l)) (/ (* t_1 t) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_1 * t) / 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 = sin(k) ** 2.0d0
    if (k <= 12500.0d0) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else if (k <= 9.2d+157) then
        tmp = 2.0d0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l)
    else
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((t_1 * t) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else if (k <= 9.2e+157) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (Math.cos(k) * l)) / l);
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_1 * t) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 12500.0:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	elif k <= 9.2e+157:
		tmp = 2.0 / (((((k * k) * t) * t_1) / (math.cos(k) * l)) / l)
	else:
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_1 * t) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 12500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	elseif (k <= 9.2e+157)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * t_1) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(t_1 * t) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 12500.0)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	elseif (k <= 9.2e+157)
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	else
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_1 * t) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 12500.0], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.2e+157], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * t), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 12500

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 12500 < k < 9.20000000000000015e157

   1. Initial program 48.2%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 84.3%

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

   if 9.20000000000000015e157 < k

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 92.1

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

   7. Applied rewrites 92.1%

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

Alternative 13: 71.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 12500:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t\_1}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \frac{t\_1 \cdot t}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 12500.0)
     (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
     (if (<= k 2.15e+160)
       (/ 2.0 (/ (/ (* (* (* k k) t) t_1) (* (cos k) l)) l))
       (/ 2.0 (* (* k (/ k (* l l))) (/ (* t_1 t) (cos k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else if (k <= 2.15e+160) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((k * (k / (l * l))) * ((t_1 * t) / 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 = sin(k) ** 2.0d0
    if (k <= 12500.0d0) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else if (k <= 2.15d+160) then
        tmp = 2.0d0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l)
    else
        tmp = 2.0d0 / ((k * (k / (l * l))) * ((t_1 * t) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 12500.0) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else if (k <= 2.15e+160) {
		tmp = 2.0 / (((((k * k) * t) * t_1) / (Math.cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((k * (k / (l * l))) * ((t_1 * t) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 12500.0:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	elif k <= 2.15e+160:
		tmp = 2.0 / (((((k * k) * t) * t_1) / (math.cos(k) * l)) / l)
	else:
		tmp = 2.0 / ((k * (k / (l * l))) * ((t_1 * t) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 12500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	elseif (k <= 2.15e+160)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * t_1) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k / Float64(l * l))) * Float64(Float64(t_1 * t) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 12500.0)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	elseif (k <= 2.15e+160)
		tmp = 2.0 / (((((k * k) * t) * t_1) / (cos(k) * l)) / l);
	else
		tmp = 2.0 / ((k * (k / (l * l))) * ((t_1 * t) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 12500.0], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.15e+160], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * t), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 12500

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 12500 < k < 2.14999999999999994e160

   1. Initial program 48.1%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 72.4

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

   5. Applied rewrites 72.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 83.8%

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

   if 2.14999999999999994e160 < k

   1. Initial program 49.3%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 62.5

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

   5. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 14: 70.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 13500:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 13500.0)
   (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
   (/ 2.0 (/ (* (* k (* k t)) (pow (sin k) 2.0)) (* (* (cos k) l) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 13500.0) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((k * (k * t)) * pow(sin(k), 2.0)) / ((cos(k) * l) * l));
	}
	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 (k <= 13500.0d0) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else
        tmp = 2.0d0 / (((k * (k * t)) * (sin(k) ** 2.0d0)) / ((cos(k) * l) * l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 13500.0) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((k * (k * t)) * Math.pow(Math.sin(k), 2.0)) / ((Math.cos(k) * l) * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 13500.0:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	else:
		tmp = 2.0 / (((k * (k * t)) * math.pow(math.sin(k), 2.0)) / ((math.cos(k) * l) * l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 13500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t)) * (sin(k) ^ 2.0)) / Float64(Float64(cos(k) * l) * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 13500.0)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	else
		tmp = 2.0 / (((k * (k * t)) * (sin(k) ^ 2.0)) / ((cos(k) * l) * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 13500.0], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 13500

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 13500 < k

   1. Initial program 48.6%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-cos.f64 67.6

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

   5. Applied rewrites 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.1

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

   9. Applied rewrites 76.1%

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

Alternative 15: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+297}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+297)
   (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
   (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k)) (tan k)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+297) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k)) * tan(k)) * 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 ((l * l) <= 1d+297) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k)) * tan(k)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+297) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+297:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	else:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k)) * math.tan(k)) * 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+297)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k)) * tan(k)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+297)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	else
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k)) * tan(k)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+297], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e297

   1. Initial program 62.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 70.6

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

   8. Applied rewrites 70.6%

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

   if 1e297 < (*.f64 l l)

   1. Initial program 33.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 61.0

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

   7. Applied rewrites 61.0%

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

Alternative 16: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 13500:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 13500.0)
   (/ 2.0 (* (/ (* (pow (* k t) 2.0) 2.0) (* (cos k) (* l l))) t))
   (/
    2.0
    (/
     (* (* (* k k) t) (- 0.5 (* 0.5 (cos (* 2.0 k)))))
     (* (* (cos k) l) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 13500.0) {
		tmp = 2.0 / (((pow((k * t), 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / ((((k * k) * t) * (0.5 - (0.5 * cos((2.0 * k))))) / ((cos(k) * l) * l));
	}
	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 (k <= 13500.0d0) then
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) * 2.0d0) / (cos(k) * (l * l))) * t)
    else
        tmp = 2.0d0 / ((((k * k) * t) * (0.5d0 - (0.5d0 * cos((2.0d0 * k))))) / ((cos(k) * l) * l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 13500.0) {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) * 2.0) / (Math.cos(k) * (l * l))) * t);
	} else {
		tmp = 2.0 / ((((k * k) * t) * (0.5 - (0.5 * Math.cos((2.0 * k))))) / ((Math.cos(k) * l) * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 13500.0:
		tmp = 2.0 / (((math.pow((k * t), 2.0) * 2.0) / (math.cos(k) * (l * l))) * t)
	else:
		tmp = 2.0 / ((((k * k) * t) * (0.5 - (0.5 * math.cos((2.0 * k))))) / ((math.cos(k) * l) * l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 13500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))) / Float64(Float64(cos(k) * l) * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 13500.0)
		tmp = 2.0 / (((((k * t) ^ 2.0) * 2.0) / (cos(k) * (l * l))) * t);
	else
		tmp = 2.0 / ((((k * k) * t) * (0.5 - (0.5 * cos((2.0 * k))))) / ((cos(k) * l) * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 13500.0], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 13500

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 13500 < k

   1. Initial program 48.6%

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

   3. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 67.6

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot \color{blue}{t}}{\cos k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{\cos k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2}} \cdot t}{\cos k}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2}} \cdot t}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot \color{blue}{t}}{\cos k}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. frac-times N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   7. Applied rewrites 70.1%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\sin k \cdot \sin k\right)}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      8. lower-*.f64 70.0

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 70.0%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.4e-21)
   (/ 2.0 (* (/ (* k k) (* l l)) (/ (* (* k k) t) (cos k))))
   (/ 2.0 (* (* (/ (pow (* k t) 2.0) (* l l)) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.4e-21) {
		tmp = 2.0 / (((k * k) / (l * l)) * (((k * k) * t) / cos(k)));
	} else {
		tmp = 2.0 / (((pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.4d-21) then
        tmp = 2.0d0 / (((k * k) / (l * l)) * (((k * k) * t) / cos(k)))
    else
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) / (l * l)) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.4e-21) {
		tmp = 2.0 / (((k * k) / (l * l)) * (((k * k) * t) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.4e-21:
		tmp = 2.0 / (((k * k) / (l * l)) * (((k * k) * t) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.pow((k * t), 2.0) / (l * l)) * 2.0) * t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.4e-21)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(Float64(Float64(k * k) * t) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.4e-21)
		tmp = 2.0 / (((k * k) / (l * l)) * (((k * k) * t) / cos(k)));
	else
		tmp = 2.0 / (((((k * t) ^ 2.0) / (l * l)) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.4e-21], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.4e-21

   1. Initial program 50.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 63.7

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 63.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{k}^{2} \cdot t}{\cos k}} \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\cos k}} \]

      2. lift-*.f64 58.1

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\cos k}} \]

   8. Applied rewrites 58.1%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\left(k \cdot k\right) \cdot t}{\cos k}} \]

   if 3.4e-21 < t

   1. Initial program 66.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 75.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      8. lift-*.f64 71.4

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   8. Applied rewrites 71.4%

      \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 60.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.55e-29)
   (/ 2.0 (/ (* (* (* k k) t) (* k k)) (* (* (cos k) l) l)))
   (/ 2.0 (* (* (/ (pow (* k t) 2.0) (* l l)) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-29) {
		tmp = 2.0 / ((((k * k) * t) * (k * k)) / ((cos(k) * l) * l));
	} else {
		tmp = 2.0 / (((pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.55d-29) then
        tmp = 2.0d0 / ((((k * k) * t) * (k * k)) / ((cos(k) * l) * l))
    else
        tmp = 2.0d0 / (((((k * t) ** 2.0d0) / (l * l)) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-29) {
		tmp = 2.0 / ((((k * k) * t) * (k * k)) / ((Math.cos(k) * l) * l));
	} else {
		tmp = 2.0 / (((Math.pow((k * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.55e-29:
		tmp = 2.0 / ((((k * k) * t) * (k * k)) / ((math.cos(k) * l) * l))
	else:
		tmp = 2.0 / (((math.pow((k * t), 2.0) / (l * l)) * 2.0) * t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.55e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) * Float64(k * k)) / Float64(Float64(cos(k) * l) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.55e-29)
		tmp = 2.0 / ((((k * k) * t) * (k * k)) / ((cos(k) * l) * l));
	else
		tmp = 2.0 / (((((k * t) ^ 2.0) / (l * l)) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.55e-29], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.55000000000000013e-29

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 63.6

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 63.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot \color{blue}{t}}{\cos k}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{\cos k}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2}} \cdot t}{\cos k}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{\ell \cdot \ell} \cdot \frac{\color{blue}{{\sin k}^{2}} \cdot t}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot \color{blue}{t}}{\cos k}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. frac-times N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

   7. Applied rewrites 64.6%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {k}^{2}}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      2. lift-*.f64 56.7

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   10. Applied rewrites 56.7%

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}{\left(\cos k \cdot \color{blue}{\ell}\right) \cdot \ell}} \]

   if 1.55000000000000013e-29 < t

   1. Initial program 67.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 75.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      8. lift-*.f64 71.2

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   8. Applied rewrites 71.2%

      \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 62.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.45e-22)
   (/ 2.0 (* (pow (/ k l) 2.0) (* (* k k) t)))
   (/ (* l l) (* (pow (* k t) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.45e-22) {
		tmp = 2.0 / (pow((k / l), 2.0) * ((k * k) * t));
	} else {
		tmp = (l * l) / (pow((k * t), 2.0) * t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.45d-22) then
        tmp = 2.0d0 / (((k / l) ** 2.0d0) * ((k * k) * t))
    else
        tmp = (l * l) / (((k * t) ** 2.0d0) * t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.45e-22) {
		tmp = 2.0 / (Math.pow((k / l), 2.0) * ((k * k) * t));
	} else {
		tmp = (l * l) / (Math.pow((k * t), 2.0) * t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.45e-22:
		tmp = 2.0 / (math.pow((k / l), 2.0) * ((k * k) * t))
	else:
		tmp = (l * l) / (math.pow((k * t), 2.0) * t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.45e-22)
		tmp = Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64(Float64(k * k) * t)));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k * t) ^ 2.0) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.45e-22)
		tmp = 2.0 / (((k / l) ^ 2.0) * ((k * k) * t));
	else
		tmp = (l * l) / (((k * t) ^ 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.45e-22], N[(2.0 / N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-22

   1. Initial program 50.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 63.7

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 63.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 55.6

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   8. Applied rewrites 55.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot \color{blue}{k}\right) \cdot t\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(\color{blue}{k} \cdot k\right) \cdot t\right)} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

      7. lift-/.f64 58.8

         \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\left(\color{blue}{k} \cdot k\right) \cdot t\right)} \]

   10. Applied rewrites 58.8%

       \[\leadsto \color{blue}{\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

   if 1.4500000000000001e-22 < t

   1. Initial program 67.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 56.3

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 56.3

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   7. Applied rewrites 56.3%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 71.9

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 71.9%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 55.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.55e-29)
   (/
    2.0
    (*
     (/
      (*
       (fma
        (+
         (fma
          (-
           (fma
            (fma -0.006349206349206349 (* t t) 0.044444444444444446)
            (* k k)
            (* 0.08888888888888889 (* t t)))
           0.3333333333333333)
          (* k k)
          (* -0.6666666666666666 (* t t)))
         1.0)
        (* k k)
        (* (* t t) 2.0))
       (* k k))
      (* 1.0 (* l l)))
     t))
   (/ (* l l) (* (pow (* k t) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-29) {
		tmp = 2.0 / (((fma((fma((fma(fma(-0.006349206349206349, (t * t), 0.044444444444444446), (k * k), (0.08888888888888889 * (t * t))) - 0.3333333333333333), (k * k), (-0.6666666666666666 * (t * t))) + 1.0), (k * k), ((t * t) * 2.0)) * (k * k)) / (1.0 * (l * l))) * t);
	} else {
		tmp = (l * l) / (pow((k * t), 2.0) * t);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.55e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(Float64(fma(fma(-0.006349206349206349, Float64(t * t), 0.044444444444444446), Float64(k * k), Float64(0.08888888888888889 * Float64(t * t))) - 0.3333333333333333), Float64(k * k), Float64(-0.6666666666666666 * Float64(t * t))) + 1.0), Float64(k * k), Float64(Float64(t * t) * 2.0)) * Float64(k * k)) / Float64(1.0 * Float64(l * l))) * t));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k * t) ^ 2.0) * t));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.55e-29], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.006349206349206349 * N[(t * t), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(0.08888888888888889 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(-0.6666666666666666 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.55000000000000013e-29

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.8%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{1 \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 1.55000000000000013e-29 < t

   1. Initial program 67.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 56.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 56.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   7. Applied rewrites 56.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 71.6

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 71.6%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 63.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t}\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, -0.006349206349206349, 0.08888888888888889\right) \cdot \left(k \cdot k\right) - 0.6666666666666666, k \cdot k, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.6e-153)
   (/ (* l l) (* (pow (* k t) 2.0) t))
   (if (<= k 2.55e+33)
     (/
      2.0
      (*
       (/
        (*
         (*
          (fma
           (-
            (* (fma (* k k) -0.006349206349206349 0.08888888888888889) (* k k))
            0.6666666666666666)
           (* k k)
           2.0)
          (* t t))
         (* k k))
        (* (fma (- (* 0.041666666666666664 (* k k)) 0.5) (* k k) 1.0) (* l l)))
       t))
     (/ 2.0 (* (/ (* k k) (* l l)) (* k (* k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-153) {
		tmp = (l * l) / (pow((k * t), 2.0) * t);
	} else if (k <= 2.55e+33) {
		tmp = 2.0 / ((((fma(((fma((k * k), -0.006349206349206349, 0.08888888888888889) * (k * k)) - 0.6666666666666666), (k * k), 2.0) * (t * t)) * (k * k)) / (fma(((0.041666666666666664 * (k * k)) - 0.5), (k * k), 1.0) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.6e-153)
		tmp = Float64(Float64(l * l) / Float64((Float64(k * t) ^ 2.0) * t));
	elseif (k <= 2.55e+33)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(k * k), -0.006349206349206349, 0.08888888888888889) * Float64(k * k)) - 0.6666666666666666), Float64(k * k), 2.0) * Float64(t * t)) * Float64(k * k)) / Float64(fma(Float64(Float64(0.041666666666666664 * Float64(k * k)) - 0.5), Float64(k * k), 1.0) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(k * Float64(k * t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.6e-153], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.55e+33], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.041666666666666664 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.59999999999999975e-153

   1. Initial program 56.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 50.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   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. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 50.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   7. Applied rewrites 50.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 64.5

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   9. Applied rewrites 64.5%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]

   if 6.59999999999999975e-153 < k < 2.5499999999999999e33

   1. Initial program 62.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 80.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(1 + {k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right)\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left({k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) \cdot {k}^{2} + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 55.0

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 55.0%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot \left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right) \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right) \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 67.2%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, -0.006349206349206349, 0.08888888888888889\right) \cdot \left(k \cdot k\right) - 0.6666666666666666, k \cdot k, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 2.5499999999999999e33 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 67.3

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 67.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 56.4

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   8. Applied rewrites 56.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      5. lower-*.f64 56.4

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)} \]

   10. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 58.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, -0.006349206349206349, 0.08888888888888889\right) \cdot \left(k \cdot k\right) - 0.6666666666666666, k \cdot k, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.5e-165)
   (/ (* l l) (* k (* k (pow t 3.0))))
   (if (<= k 2.55e+33)
     (/
      2.0
      (*
       (/
        (*
         (*
          (fma
           (-
            (* (fma (* k k) -0.006349206349206349 0.08888888888888889) (* k k))
            0.6666666666666666)
           (* k k)
           2.0)
          (* t t))
         (* k k))
        (* (fma (- (* 0.041666666666666664 (* k k)) 0.5) (* k k) 1.0) (* l l)))
       t))
     (/ 2.0 (* (/ (* k k) (* l l)) (* k (* k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e-165) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else if (k <= 2.55e+33) {
		tmp = 2.0 / ((((fma(((fma((k * k), -0.006349206349206349, 0.08888888888888889) * (k * k)) - 0.6666666666666666), (k * k), 2.0) * (t * t)) * (k * k)) / (fma(((0.041666666666666664 * (k * k)) - 0.5), (k * k), 1.0) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.5e-165)
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (t ^ 3.0))));
	elseif (k <= 2.55e+33)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(k * k), -0.006349206349206349, 0.08888888888888889) * Float64(k * k)) - 0.6666666666666666), Float64(k * k), 2.0) * Float64(t * t)) * Float64(k * k)) / Float64(fma(Float64(Float64(0.041666666666666664 * Float64(k * k)) - 0.5), Float64(k * k), 1.0) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(k * Float64(k * t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.5e-165], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.55e+33], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.041666666666666664 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.49999999999999992e-165

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 50.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{{t}^{3}}\right)} \]

      7. lift-pow.f64 56.8

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{\color{blue}{3}}\right)} \]

   7. Applied rewrites 56.8%

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   if 4.49999999999999992e-165 < k < 2.5499999999999999e33

   1. Initial program 62.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 80.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 56.3%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(1 + {k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right)\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left({k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) \cdot {k}^{2} + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 55.3

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 55.3%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot \left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right) \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right) \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 67.6%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, -0.006349206349206349, 0.08888888888888889\right) \cdot \left(k \cdot k\right) - 0.6666666666666666, k \cdot k, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 2.5499999999999999e33 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 67.3

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 67.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 56.4

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   8. Applied rewrites 56.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      5. lower-*.f64 56.4

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)} \]

   10. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 46.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, -0.006349206349206349, 0.08888888888888889\right) \cdot \left(k \cdot k\right) - 0.6666666666666666, k \cdot k, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.55e+33)
   (/
    2.0
    (*
     (/
      (*
       (*
        (fma
         (-
          (* (fma (* k k) -0.006349206349206349 0.08888888888888889) (* k k))
          0.6666666666666666)
         (* k k)
         2.0)
        (* t t))
       (* k k))
      (* (fma (- (* 0.041666666666666664 (* k k)) 0.5) (* k k) 1.0) (* l l)))
     t))
   (/ 2.0 (* (/ (* k k) (* l l)) (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.55e+33) {
		tmp = 2.0 / ((((fma(((fma((k * k), -0.006349206349206349, 0.08888888888888889) * (k * k)) - 0.6666666666666666), (k * k), 2.0) * (t * t)) * (k * k)) / (fma(((0.041666666666666664 * (k * k)) - 0.5), (k * k), 1.0) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.55e+33)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(k * k), -0.006349206349206349, 0.08888888888888889) * Float64(k * k)) - 0.6666666666666666), Float64(k * k), 2.0) * Float64(t * t)) * Float64(k * k)) / Float64(fma(Float64(Float64(0.041666666666666664 * Float64(k * k)) - 0.5), Float64(k * k), 1.0) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(k * Float64(k * t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.55e+33], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.006349206349206349 + 0.08888888888888889), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.041666666666666664 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5499999999999999e33

   1. Initial program 57.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 75.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 48.0%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(1 + {k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right)\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left({k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) \cdot {k}^{2} + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 38.2

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 38.2%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot \left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right) \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(2 + {k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{4}{45} + \frac{-2}{315} \cdot {k}^{2}\right) - \frac{2}{3}\right)\right) \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 43.7%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(k \cdot k, -0.006349206349206349, 0.08888888888888889\right) \cdot \left(k \cdot k\right) - 0.6666666666666666, k \cdot k, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 2.5499999999999999e33 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 67.3

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 67.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 56.4

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   8. Applied rewrites 56.4%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      5. lower-*.f64 56.4

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)} \]

   10. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 47.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 26500:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 26500.0)
   (/
    2.0
    (*
     (/
      (* (* (* t t) 2.0) (* k k))
      (* (fma (- (* 0.041666666666666664 (* k k)) 0.5) (* k k) 1.0) (* l l)))
     t))
   (/ 2.0 (* (/ (* k k) (* l l)) (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 26500.0) {
		tmp = 2.0 / (((((t * t) * 2.0) * (k * k)) / (fma(((0.041666666666666664 * (k * k)) - 0.5), (k * k), 1.0) * (l * l))) * t);
	} else {
		tmp = 2.0 / (((k * k) / (l * l)) * (k * (k * t)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 26500.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * t) * 2.0) * Float64(k * k)) / Float64(fma(Float64(Float64(0.041666666666666664 * Float64(k * k)) - 0.5), Float64(k * k), 1.0) * Float64(l * l))) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l * l)) * Float64(k * Float64(k * t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 26500.0], N[(2.0 / N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.041666666666666664 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 26500

   1. Initial program 57.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 48.6%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(1 + {k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right)\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left({k}^{2} \cdot \left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\left(\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}\right) \cdot {k}^{2} + 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot {k}^{2} - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, {k}^{2}, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{315}, t \cdot t, \frac{2}{45}\right), k \cdot k, \frac{4}{45} \cdot \left(t \cdot t\right)\right) - \frac{1}{3}, k \cdot k, \frac{-2}{3} \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 38.6

          \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 38.6%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2}\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left({t}^{2} \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(\frac{1}{24} \cdot \left(k \cdot k\right) - \frac{1}{2}, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. lift-*.f64 44.6

          \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   14. Applied rewrites 44.6%

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(k \cdot k\right) - 0.5, k \cdot k, 1\right) \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   if 26500 < k

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{\cos k}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{{\ell}^{2}} \cdot \frac{\color{blue}{t} \cdot {\sin k}^{2}}{\cos k}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot \color{blue}{{\sin k}^{2}}}{\cos k}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\cos k}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos \color{blue}{k}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

      13. lower-cos.f64 67.6

          \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}} \]

   5. Applied rewrites 67.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2} \cdot t}{\cos k}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot t\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. lift-*.f64 54.9

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   8. Applied rewrites 54.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

      5. lower-*.f64 54.9

         \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)} \]

   10. Applied rewrites 54.9%

       \[\leadsto \frac{2}{\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 51.2% 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 55.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 51.2

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
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. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. lift-*.f64 51.2

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

7. Applied rewrites 51.2%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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))))