Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.8% → 90.7%

Time: 8.5s

Alternatives: 11

Speedup: 7.7×

• 39.9% 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: 34.8% 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: 90.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l / (sqrt(k_m) * sqrt(k_m))) * (l / k_m)) * ((cos(k_m) * 2.0d0) / (t * (sin(k_m) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

2. Taylor expanded in t around 0

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

   1. add-sqr-sqrt N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 49.7%

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

5. Taylor expanded in t around 0

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

   1. times-frac N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

7. Applied rewrites 90.8%

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

8. Taylor expanded in k around 0

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

   1. add-sqr-sqrt N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 90.7

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

10. Applied rewrites 90.7%

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

Alternative 2: 70.8% accurate, 0.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (- (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      -1e-10)
   (/
    2.0
    (*
     (* (* (/ (* (* t t) t) (* l l)) (sin k_m)) (tan k_m))
     (/ (* k_m k_m) (* t t))))
   (*
    (* (/ l (* (sqrt k_m) (sqrt k_m))) (/ l k_m))
    (/ 2.0 (* t (pow (sin k_m) 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) - 1.0))) <= -1e-10) {
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * sin(k_m)) * tan(k_m)) * ((k_m * k_m) / (t * t)));
	} else {
		tmp = ((l / (sqrt(k_m) * sqrt(k_m))) * (l / k_m)) * (2.0 / (t * pow(sin(k_m), 2.0)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) - 1.0d0))) <= (-1d-10)) then
        tmp = 2.0d0 / ((((((t * t) * t) / (l * l)) * sin(k_m)) * tan(k_m)) * ((k_m * k_m) / (t * t)))
    else
        tmp = ((l / (sqrt(k_m) * sqrt(k_m))) * (l / k_m)) * (2.0d0 / (t * (sin(k_m) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) - 1.0))) <= -1e-10) {
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((k_m * k_m) / (t * t)));
	} else {
		tmp = ((l / (Math.sqrt(k_m) * Math.sqrt(k_m))) * (l / k_m)) * (2.0 / (t * Math.pow(Math.sin(k_m), 2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) - 1.0))) <= -1e-10:
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((k_m * k_m) / (t * t)))
	else:
		tmp = ((l / (math.sqrt(k_m) * math.sqrt(k_m))) * (l / k_m)) * (2.0 / (t * math.pow(math.sin(k_m), 2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) - 1.0))) <= -1e-10)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(k_m * k_m) / Float64(t * t))));
	else
		tmp = Float64(Float64(Float64(l / Float64(sqrt(k_m) * sqrt(k_m))) * Float64(l / k_m)) * Float64(2.0 / Float64(t * (sin(k_m) ^ 2.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) - 1.0))) <= -1e-10)
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * sin(k_m)) * tan(k_m)) * ((k_m * k_m) / (t * t)));
	else
		tmp = ((l / (sqrt(k_m) * sqrt(k_m))) * (l / k_m)) * (2.0 / (t * (sin(k_m) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[Sqrt[k$95$m], $MachinePrecision] * N[Sqrt[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $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)))) < -1.00000000000000004e-10

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 66.4

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

   6. Applied rewrites 66.4%

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

   if -1.00000000000000004e-10 < (/.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 27.2%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in k around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 89.5

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

   10. Applied rewrites 89.5%

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

   11. Taylor expanded in k around 0

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

       1. add-sqr-sqrt 71.4

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

   13. Applied rewrites 71.4%

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

Alternative 3: 70.8% accurate, 0.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (- (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      -1e-10)
   (/
    2.0
    (*
     (* (* (/ (* (* t t) t) (* l l)) (sin k_m)) (tan k_m))
     (/ (* k_m k_m) (* t t))))
   (* (* (/ l k_m) (/ l k_m)) (/ 2.0 (* t (pow (sin k_m) 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) - 1.0))) <= -1e-10) {
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * sin(k_m)) * tan(k_m)) * ((k_m * k_m) / (t * t)));
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t * pow(sin(k_m), 2.0)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) - 1.0d0))) <= (-1d-10)) then
        tmp = 2.0d0 / ((((((t * t) * t) / (l * l)) * sin(k_m)) * tan(k_m)) * ((k_m * k_m) / (t * t)))
    else
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 / (t * (sin(k_m) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) - 1.0))) <= -1e-10) {
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((k_m * k_m) / (t * t)));
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t * Math.pow(Math.sin(k_m), 2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) - 1.0))) <= -1e-10:
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((k_m * k_m) / (t * t)))
	else:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t * math.pow(math.sin(k_m), 2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) - 1.0))) <= -1e-10)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) * t) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(k_m * k_m) / Float64(t * t))));
	else
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 / Float64(t * (sin(k_m) ^ 2.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) - 1.0))) <= -1e-10)
		tmp = 2.0 / ((((((t * t) * t) / (l * l)) * sin(k_m)) * tan(k_m)) * ((k_m * k_m) / (t * t)));
	else
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t * (sin(k_m) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $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)))) < -1.00000000000000004e-10

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 66.4

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

   6. Applied rewrites 66.4%

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

   if -1.00000000000000004e-10 < (/.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 27.2%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 71.4%

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

Alternative 4: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) - 1\right)} \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k\_m \cdot k\_m}{t}, 2 \cdot {t}^{-1}\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{2}{t \cdot {\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ l k_m) (/ l k_m))))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
          (- (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
        -1e-10)
     (*
      t_1
      (/
       (fma -0.3333333333333333 (/ (* k_m k_m) t) (* 2.0 (pow t -1.0)))
       (* k_m k_m)))
     (* t_1 (/ 2.0 (* t (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (l / k_m) * (l / k_m);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) - 1.0))) <= -1e-10) {
		tmp = t_1 * (fma(-0.3333333333333333, ((k_m * k_m) / t), (2.0 * pow(t, -1.0))) / (k_m * k_m));
	} else {
		tmp = t_1 * (2.0 / (t * pow(sin(k_m), 2.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(l / k_m) * Float64(l / k_m))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) - 1.0))) <= -1e-10)
		tmp = Float64(t_1 * Float64(fma(-0.3333333333333333, Float64(Float64(k_m * k_m) / t), Float64(2.0 * (t ^ -1.0))) / Float64(k_m * k_m)));
	else
		tmp = Float64(t_1 * Float64(2.0 / Float64(t * (sin(k_m) ^ 2.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-10], N[(t$95$1 * N[(N[(-0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] + N[(2.0 * N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $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)))) < -1.00000000000000004e-10

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 88.0

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, 2 \cdot {t}^{-1}\right)}{k \cdot k} \]

   10. Applied rewrites 88.0%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, 2 \cdot {t}^{-1}\right)}{\color{blue}{k \cdot k}} \]

   if -1.00000000000000004e-10 < (/.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 27.2%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 71.4%

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

Alternative 5: 80.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k\_m \cdot k\_m}{t}, 2 \cdot {t}^{-1}\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.2e-15)
   (*
    (* (/ l k_m) (/ l k_m))
    (/
     (fma -0.3333333333333333 (/ (* k_m k_m) t) (* 2.0 (pow t -1.0)))
     (* k_m k_m)))
   (*
    (* (/ (* l l) (* k_m k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t)))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-15) {
		tmp = ((l / k_m) * (l / k_m)) * (fma(-0.3333333333333333, ((k_m * k_m) / t), (2.0 * pow(t, -1.0))) / (k_m * k_m));
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.2e-15)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(fma(-0.3333333333333333, Float64(Float64(k_m * k_m) / t), Float64(2.0 * (t ^ -1.0))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.2e-15], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] + N[(2.0 * N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999997e-15

   1. Initial program 40.6%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 90.2

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, 2 \cdot {t}^{-1}\right)}{k \cdot k} \]

   10. Applied rewrites 90.2%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, 2 \cdot {t}^{-1}\right)}{\color{blue}{k \cdot k}} \]

   if 1.19999999999999997e-15 < k

   1. Initial program 29.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.8%

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

Alternative 6: 90.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l / k_m) * (l / k_m)) * ((cos(k_m) * 2.0d0) / (t * (sin(k_m) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

2. Taylor expanded in t around 0

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

   1. add-sqr-sqrt N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 49.7%

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

5. Taylor expanded in t around 0

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

   1. times-frac N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

7. Applied rewrites 90.8%

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

Alternative 7: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{-160}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(\cos k\_m \cdot 2\right)}{\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 2.05e-160)
   (* (* (/ l k_m) (/ l k_m)) (/ 2.0 (* (* k_m k_m) t)))
   (/
    (* (* l l) (* (cos k_m) 2.0))
    (* (* k_m k_m) (* t (pow (sin k_m) 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 2.05e-160) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t));
	} else {
		tmp = ((l * l) * (cos(k_m) * 2.0)) / ((k_m * k_m) * (t * pow(sin(k_m), 2.0)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 2.05d-160) then
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 / ((k_m * k_m) * t))
    else
        tmp = ((l * l) * (cos(k_m) * 2.0d0)) / ((k_m * k_m) * (t * (sin(k_m) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 2.05e-160) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t));
	} else {
		tmp = ((l * l) * (Math.cos(k_m) * 2.0)) / ((k_m * k_m) * (t * Math.pow(Math.sin(k_m), 2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 2.05e-160:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t))
	else:
		tmp = ((l * l) * (math.cos(k_m) * 2.0)) / ((k_m * k_m) * (t * math.pow(math.sin(k_m), 2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 2.05e-160)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(cos(k_m) * 2.0)) / Float64(Float64(k_m * k_m) * Float64(t * (sin(k_m) ^ 2.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 2.05e-160)
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t));
	else
		tmp = ((l * l) * (cos(k_m) * 2.0)) / ((k_m * k_m) * (t * (sin(k_m) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 2.05e-160], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.05000000000000001e-160

   1. Initial program 32.5%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 72.8

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

   10. Applied rewrites 72.8%

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

   if 2.05000000000000001e-160 < l

   1. Initial program 38.7%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. add-sqr-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

   10. Applied rewrites 79.5%

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

Alternative 8: 73.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 110:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k\_m \cdot k\_m}{t}, 2 \cdot {t}^{-1}\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 110.0)
   (*
    (* (/ l k_m) (/ l k_m))
    (/
     (fma -0.3333333333333333 (/ (* k_m k_m) t) (* 2.0 (pow t -1.0)))
     (* k_m k_m)))
   (* (* (/ (* l l) (* k_m k_m)) (/ (cos k_m) (* (* k_m k_m) t))) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 110.0) {
		tmp = ((l / k_m) * (l / k_m)) * (fma(-0.3333333333333333, ((k_m * k_m) / t), (2.0 * pow(t, -1.0))) / (k_m * k_m));
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 110.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(fma(-0.3333333333333333, Float64(Float64(k_m * k_m) / t), Float64(2.0 * (t ^ -1.0))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64(Float64(k_m * k_m) * t))) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 110.0], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] + N[(2.0 * N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 110

   1. Initial program 39.5%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 89.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 89.5

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, 2 \cdot {t}^{-1}\right)}{k \cdot k} \]

   10. Applied rewrites 89.5%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, 2 \cdot {t}^{-1}\right)}{\color{blue}{k \cdot k}} \]

   if 110 < k

   1. Initial program 29.9%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 56.0

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

   7. Applied rewrites 56.0%

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

Alternative 9: 73.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \left(k\_m \cdot k\_m\right) \cdot t\\ \mathbf{if}\;k\_m \leq 1.2:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{t\_1}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (* k_m k_m) t)))
   (if (<= k_m 1.2)
     (* (* (/ l k_m) (/ l k_m)) (/ 2.0 t_1))
     (* (* (/ (* l l) (* k_m k_m)) (/ (cos k_m) t_1)) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double tmp;
	if (k_m <= 1.2) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / t_1);
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / t_1)) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k_m * k_m) * t
    if (k_m <= 1.2d0) then
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 / t_1)
    else
        tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / t_1)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double tmp;
	if (k_m <= 1.2) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / t_1);
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (Math.cos(k_m) / t_1)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m * k_m) * t
	tmp = 0
	if k_m <= 1.2:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / t_1)
	else:
		tmp = (((l * l) / (k_m * k_m)) * (math.cos(k_m) / t_1)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m * k_m) * t)
	tmp = 0.0
	if (k_m <= 1.2)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 / t_1));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / t_1)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m * k_m) * t;
	tmp = 0.0;
	if (k_m <= 1.2)
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / t_1);
	else
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / t_1)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k$95$m, 1.2], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999996

   1. Initial program 39.6%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 89.6

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

   10. Applied rewrites 89.6%

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

   if 1.19999999999999996 < k

   1. Initial program 29.9%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 56.0

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

   7. Applied rewrites 56.0%

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

Alternative 10: 71.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.4 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{-0.16666666666666666}{t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.4e+22)
   (* (* (/ l k_m) (/ l k_m)) (/ 2.0 (* (* k_m k_m) t)))
   (* (* (/ (* l l) (* k_m k_m)) (/ -0.16666666666666666 t)) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e+22) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t));
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.4d+22) then
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 / ((k_m * k_m) * t))
    else
        tmp = (((l * l) / (k_m * k_m)) * ((-0.16666666666666666d0) / t)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e+22) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t));
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.4e+22:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t))
	else:
		tmp = (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.4e+22)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(-0.16666666666666666 / t)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.4e+22)
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / ((k_m * k_m) * t));
	else
		tmp = (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.4e+22], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e22

   1. Initial program 38.5%

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

   2. Taylor expanded in t around 0

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

      1. add-sqr-sqrt N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in t around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 86.1

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

   10. Applied rewrites 86.1%

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

   if 1.4e22 < k

   1. Initial program 30.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{{k}^{2}}\right) \cdot 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

      4. pow2 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

      6. inv-pow N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, {t}^{-1}\right)}{{k}^{2}}\right) \cdot 2 \]

      7. lower-pow.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, {t}^{-1}\right)}{{k}^{2}}\right) \cdot 2 \]

      8. pow2 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, {t}^{-1}\right)}{k \cdot k}\right) \cdot 2 \]

      9. lift-*.f64 18.3

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t}, {t}^{-1}\right)}{k \cdot k}\right) \cdot 2 \]

   7. Applied rewrites 18.3%

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t}, {t}^{-1}\right)}{k \cdot k}\right) \cdot 2 \]

   8. Taylor expanded in k around inf

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\frac{-1}{6}}{t}\right) \cdot 2 \]
   9. Step-by-step derivation

      1. lower-/.f64 54.0

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{-0.16666666666666666}{t}\right) \cdot 2 \]

   10. Applied rewrites 54.0%

       \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{-0.16666666666666666}{t}\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 28.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{-0.16666666666666666}{t}\right) \cdot 2 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* (/ (* l l) (* k_m k_m)) (/ -0.16666666666666666 t)) 2.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (((l * l) / (k_m * k_m)) * ((-0.16666666666666666d0) / t)) * 2.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(-0.16666666666666666 / t)) * 2.0)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((l * l) / (k_m * k_m)) * (-0.16666666666666666 / t)) * 2.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 34.8%

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

2. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 73.9%

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

5. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{{k}^{2}}\right) \cdot 2 \]

   2. lower-fma.f64 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

   3. lower-/.f64 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{{k}^{2}}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

   4. pow2 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, \frac{1}{t}\right)}{{k}^{2}}\right) \cdot 2 \]

   6. inv-pow N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, {t}^{-1}\right)}{{k}^{2}}\right) \cdot 2 \]

   7. lower-pow.f64 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, {t}^{-1}\right)}{{k}^{2}}\right) \cdot 2 \]

   8. pow2 N/A

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(\frac{-1}{6}, \frac{k \cdot k}{t}, {t}^{-1}\right)}{k \cdot k}\right) \cdot 2 \]

   9. lift-*.f64 48.5

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t}, {t}^{-1}\right)}{k \cdot k}\right) \cdot 2 \]

7. Applied rewrites 48.5%

   \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{k \cdot k}{t}, {t}^{-1}\right)}{k \cdot k}\right) \cdot 2 \]

8. Taylor expanded in k around inf

   \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\frac{-1}{6}}{t}\right) \cdot 2 \]
9. Step-by-step derivation

   1. lower-/.f64 28.6

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{-0.16666666666666666}{t}\right) \cdot 2 \]

10. Applied rewrites 28.6%

    \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{-0.16666666666666666}{t}\right) \cdot 2 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025099 
(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))))