Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.4% → 97.4%

Time: 14.4s

Alternatives: 17

Speedup: 11.0×

• 40.6% 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: 36.4% 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: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.9

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

5. Applied rewrites 91.9%

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

7. Applied rewrites 96.4%

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

9. Applied rewrites 98.9%

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

Alternative 2: 86.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell \cdot \cos k}\\ \mathbf{if}\;k \leq 1.18 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (* l (cos k)))))
   (if (<= k 1.18e-27)
     (/
      2.0
      (*
       (* k (/ (* (* (fma (* (* k k) t) -0.3333333333333333 t) k) k) l))
       t_1))
     (/ 2.0 (* (* (pow (sin k) 2.0) (* t (/ k l))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = k / (l * cos(k));
	double tmp;
	if (k <= 1.18e-27) {
		tmp = 2.0 / ((k * (((fma(((k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * t_1);
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t * (k / l))) * t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / Float64(l * cos(k)))
	tmp = 0.0
	if (k <= 1.18e-27)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(Float64(Float64(k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t * Float64(k / l))) * t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.18e-27], N[(2.0 / N[(N[(k * N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.18e-27

   1. Initial program 36.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.3

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 79.7%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.2%

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

   if 1.18e-27 < k

   1. Initial program 30.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 99.5%

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

Alternative 3: 86.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\cos k \cdot \ell} \cdot \left({\sin k}^{2} \cdot t\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1e-5)
   (/
    2.0
    (*
     (* k (/ (* (* (fma (* (* k k) t) -0.3333333333333333 t) k) k) l))
     (/ k (* l (cos k)))))
   (/ 2.0 (* (* (/ k (* (cos k) l)) (* (pow (sin k) 2.0) t)) (/ k l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1e-5) {
		tmp = 2.0 / ((k * (((fma(((k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * (k / (l * cos(k))));
	} else {
		tmp = 2.0 / (((k / (cos(k) * l)) * (pow(sin(k), 2.0) * t)) * (k / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1e-5)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(Float64(Float64(k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * Float64(k / Float64(l * cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / Float64(cos(k) * l)) * Float64((sin(k) ^ 2.0) * t)) * Float64(k / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1e-5], N[(2.0 / N[(N[(k * N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000008e-5

   1. Initial program 36.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 80.1%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.5%

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

   if 1.00000000000000008e-5 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 99.3%

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

Alternative 4: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell \cdot \cos k}\\ \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ k (* l (cos k)))))
   (if (<= k 0.0025)
     (/
      2.0
      (*
       (* k (/ (* (* (fma (* (* k k) t) -0.3333333333333333 t) k) k) l))
       t_1))
     (/ 2.0 (* (* (- 0.5 (* 0.5 (cos (+ k k)))) (* t (/ k l))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = k / (l * cos(k));
	double tmp;
	if (k <= 0.0025) {
		tmp = 2.0 / ((k * (((fma(((k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * t_1);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((k + k)))) * (t * (k / l))) * t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / Float64(l * cos(k)))
	tmp = 0.0
	if (k <= 0.0025)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(Float64(Float64(k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * Float64(t * Float64(k / l))) * t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.0025], N[(2.0 / N[(N[(k * N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 36.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 80.1%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.5%

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

   if 0.00250000000000000005 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 98.9%

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

Alternative 5: 83.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0055:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k} \cdot \left(\frac{\ell}{k} \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0055)
   (/
    2.0
    (*
     (* k (/ (* (* (fma (* (* k k) t) -0.3333333333333333 t) k) k) l))
     (/ k (* l (cos k)))))
   (*
    (/ (* 2.0 (cos k)) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) t) k))
    (* (/ l k) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0055) {
		tmp = 2.0 / ((k * (((fma(((k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * (k / (l * cos(k))));
	} else {
		tmp = ((2.0 * cos(k)) / (((0.5 - (0.5 * cos((k + k)))) * t) * k)) * ((l / k) * l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0055)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(Float64(Float64(k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * Float64(k / Float64(l * cos(k)))));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * t) * k)) * Float64(Float64(l / k) * l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.0055], N[(2.0 / N[(N[(k * N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0054999999999999997

   1. Initial program 36.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 80.1%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.5%

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

   if 0.0054999999999999997 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 89.0%

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

   9. Applied rewrites 88.6%

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

Alternative 6: 80.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0055:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos k \cdot 2\right) \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot k\right) \cdot k}}{\mathsf{fma}\left(\cos \left(-2 \cdot k\right), -0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0055)
   (/
    2.0
    (*
     (* k (/ (* (* (fma (* (* k k) t) -0.3333333333333333 t) k) k) l))
     (/ k (* l (cos k)))))
   (/
    (/ (* (* (cos k) 2.0) (* l l)) (* (* t k) k))
    (fma (cos (* -2.0 k)) -0.5 0.5))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0055) {
		tmp = 2.0 / ((k * (((fma(((k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * (k / (l * cos(k))));
	} else {
		tmp = (((cos(k) * 2.0) * (l * l)) / ((t * k) * k)) / fma(cos((-2.0 * k)), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0055)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(Float64(Float64(k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * Float64(k / Float64(l * cos(k)))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * 2.0) * Float64(l * l)) / Float64(Float64(t * k) * k)) / fma(cos(Float64(-2.0 * k)), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.0055], N[(2.0 / N[(N[(k * N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * 2.0), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(-2.0 * k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0054999999999999997

   1. Initial program 36.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 80.1%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.5%

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

   if 0.0054999999999999997 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 77.8%

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

Alternative 7: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot t\right) \cdot k} \cdot \left(\frac{\ell}{k} \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9e+14)
   (/
    2.0
    (*
     (* k (/ (* (* (fma (* (* k k) t) -0.3333333333333333 t) k) k) l))
     (/ k (* l (cos k)))))
   (* (/ 2.0 (* (* (pow (sin k) 2.0) t) k)) (* (/ l k) l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e+14) {
		tmp = 2.0 / ((k * (((fma(((k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * (k / (l * cos(k))));
	} else {
		tmp = (2.0 / ((pow(sin(k), 2.0) * t) * k)) * ((l / k) * l);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9e+14)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(Float64(Float64(k * k) * t), -0.3333333333333333, t) * k) * k) / l)) * Float64(k / Float64(l * cos(k)))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * t) * k)) * Float64(Float64(l / k) * l));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9e+14], N[(2.0 / N[(N[(k * N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9e14

   1. Initial program 36.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 79.7%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.0%

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

   if 9e14 < k

   1. Initial program 31.0%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 51.3%

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

Alternative 8: 74.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(k \cdot k\right) \cdot t\\ \mathbf{if}\;k \leq 1.55:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\left(\mathsf{fma}\left(t\_1, -0.3333333333333333, t\right) \cdot k\right) \cdot k}{\ell}\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{t\_1 \cdot k} \cdot \frac{\ell \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* k k) t)))
   (if (<= k 1.55)
     (/
      2.0
      (*
       (* k (/ (* (* (fma t_1 -0.3333333333333333 t) k) k) l))
       (/ k (* l (cos k)))))
     (* (/ (* 2.0 (cos k)) (* t_1 k)) (/ (* l l) k)))))

C

double code(double t, double l, double k) {
	double t_1 = (k * k) * t;
	double tmp;
	if (k <= 1.55) {
		tmp = 2.0 / ((k * (((fma(t_1, -0.3333333333333333, t) * k) * k) / l)) * (k / (l * cos(k))));
	} else {
		tmp = ((2.0 * cos(k)) / (t_1 * k)) * ((l * l) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k * k) * t)
	tmp = 0.0
	if (k <= 1.55)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(Float64(fma(t_1, -0.3333333333333333, t) * k) * k) / l)) * Float64(k / Float64(l * cos(k)))));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(t_1 * k)) * Float64(Float64(l * l) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k, 1.55], N[(2.0 / N[(N[(k * N[(N[(N[(N[(t$95$1 * -0.3333333333333333 + t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 80.2%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.6%

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

   if 1.55000000000000004 < k

   1. Initial program 30.1%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 53.9%

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

Alternative 9: 70.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left({k}^{4} \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 6.8e-162)
   (*
    (/ (* 2.0 l) (* (* (pow k 4.0) t) (fma (* k k) 0.16666666666666666 1.0)))
    l)
   (* (/ (* 2.0 (cos k)) (* (* (* k k) t) k)) (/ (* l l) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.8e-162) {
		tmp = ((2.0 * l) / ((pow(k, 4.0) * t) * fma((k * k), 0.16666666666666666, 1.0))) * l;
	} else {
		tmp = ((2.0 * cos(k)) / (((k * k) * t) * k)) * ((l * l) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 6.8e-162)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(Float64((k ^ 4.0) * t) * fma(Float64(k * k), 0.16666666666666666, 1.0))) * l);
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(Float64(k * k) * t) * k)) * Float64(Float64(l * l) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 6.8e-162], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.8e-162

   1. Initial program 37.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + \frac{1}{6} \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t + \frac{1}{6} \cdot \left({k}^{2} \cdot t\right)\right) \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{1}{6}}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{{k}^{2} \cdot \left(t \cdot \frac{1}{6}\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + {k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot t\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t + {k}^{2} \cdot \left(\frac{1}{6} \cdot t\right)\right) \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + {k}^{2} \cdot \color{blue}{\left(t \cdot \frac{1}{6}\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{1}{6}}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\frac{1}{6} \cdot \left({k}^{2} \cdot t\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot t}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot t\right)} \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot t\right)} \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)} \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      15. lower-pow.f64 64.4

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot t\right) \cdot \color{blue}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied rewrites 64.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \ell\right) \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \ell\right) \cdot \ell} \]

   9. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left({k}^{4} \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right)} \cdot \ell} \]

   if 6.8e-162 < l

   1. Initial program 32.0%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 63.1%

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

Alternative 10: 75.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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 / (((k * k) * (t * (k / l))) * (k / (l * cos(k))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.9

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

5. Applied rewrites 91.9%

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 73.9%

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

Alternative 11: 71.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right) \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-51)
   (* (/ l (* (* t (* k k)) (* k k))) (* 2.0 l))
   (if (<= k 1.55e+63)
     (* (fma (* k k) -0.3333333333333333 2.0) (* (/ l (pow k 4.0)) (/ l t)))
     (* (/ 2.0 (* (* (* k k) t) k)) (/ (* l l) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-51) {
		tmp = (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
	} else if (k <= 1.55e+63) {
		tmp = fma((k * k), -0.3333333333333333, 2.0) * ((l / pow(k, 4.0)) * (l / t));
	} else {
		tmp = (2.0 / (((k * k) * t) * k)) * ((l * l) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-51)
		tmp = Float64(Float64(l / Float64(Float64(t * Float64(k * k)) * Float64(k * k))) * Float64(2.0 * l));
	elseif (k <= 1.55e+63)
		tmp = Float64(fma(Float64(k * k), -0.3333333333333333, 2.0) * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(k * k) * t) * k)) * Float64(Float64(l * l) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.2e-51], N[(N[(l / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+63], N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.1999999999999995e-51

   1. Initial program 36.0%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{{k}^{4} \cdot t}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{4} \cdot t} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      5. associate-/l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{4} \cdot t} + \color{blue}{\ell \cdot \frac{\ell}{{k}^{4} \cdot t}} \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t}} \cdot \left(\ell + \ell\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell + \ell\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{{k}^{4}} \cdot t} \cdot \left(\ell + \ell\right) \]

      11. count-2-rev N/A

          \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

      12. lower-*.f64 74.5

          \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 77.2%

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

   if 6.1999999999999995e-51 < k < 1.55e63

   1. Initial program 40.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-/r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 80.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right) \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

   if 1.55e63 < k

   1. Initial program 28.5%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 45.9%

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

   10. Applied rewrites 47.9%

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

Alternative 12: 69.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left({k}^{4} \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 6.8e-162)
   (*
    (/ (* 2.0 l) (* (* (pow k 4.0) t) (fma (* k k) 0.16666666666666666 1.0)))
    l)
   (* (/ 2.0 (* (* t k) (* k k))) (/ (* l l) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.8e-162) {
		tmp = ((2.0 * l) / ((pow(k, 4.0) * t) * fma((k * k), 0.16666666666666666, 1.0))) * l;
	} else {
		tmp = (2.0 / ((t * k) * (k * k))) * ((l * l) / k);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 6.8e-162)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(Float64((k ^ 4.0) * t) * fma(Float64(k * k), 0.16666666666666666, 1.0))) * l);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t * k) * Float64(k * k))) * Float64(Float64(l * l) / k));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 6.8e-162], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 / N[(N[(t * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.8e-162

   1. Initial program 37.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + \frac{1}{6} \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t + \frac{1}{6} \cdot \left({k}^{2} \cdot t\right)\right) \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{1}{6}}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{{k}^{2} \cdot \left(t \cdot \frac{1}{6}\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + {k}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot t\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t + {k}^{2} \cdot \left(\frac{1}{6} \cdot t\right)\right) \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + {k}^{2} \cdot \color{blue}{\left(t \cdot \frac{1}{6}\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{1}{6}}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\frac{1}{6} \cdot \left({k}^{2} \cdot t\right)}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t + \color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot t}\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot t\right)} \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot t\right)} \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)} \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right) \]

      15. lower-pow.f64 64.4

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot t\right) \cdot \color{blue}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied rewrites 64.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \left(\ell \cdot \ell\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \ell\right) \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{2}{\left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot t\right) \cdot {k}^{4}} \cdot \ell\right) \cdot \ell} \]

   9. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left({k}^{4} \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right)} \cdot \ell} \]

   if 6.8e-162 < l

   1. Initial program 32.0%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 56.1%

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

   10. Applied rewrites 60.0%

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

Alternative 13: 71.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 7e-162)
   (* (/ l (* (* t (* k k)) (* k k))) (* 2.0 l))
   (* (/ 2.0 (* (* t k) (* k k))) (/ (* l l) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 7e-162) {
		tmp = (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
	} else {
		tmp = (2.0 / ((t * k) * (k * k))) * ((l * l) / k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 7d-162) then
        tmp = (l / ((t * (k * k)) * (k * k))) * (2.0d0 * l)
    else
        tmp = (2.0d0 / ((t * k) * (k * k))) * ((l * l) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 7e-162) {
		tmp = (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
	} else {
		tmp = (2.0 / ((t * k) * (k * k))) * ((l * l) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 7e-162:
		tmp = (l / ((t * (k * k)) * (k * k))) * (2.0 * l)
	else:
		tmp = (2.0 / ((t * k) * (k * k))) * ((l * l) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 7e-162)
		tmp = Float64(Float64(l / Float64(Float64(t * Float64(k * k)) * Float64(k * k))) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t * k) * Float64(k * k))) * Float64(Float64(l * l) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 7e-162)
		tmp = (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
	else
		tmp = (2.0 / ((t * k) * (k * k))) * ((l * l) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 7e-162], N[(N[(l / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.9999999999999998e-162

   1. Initial program 37.1%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{{k}^{4} \cdot t}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{4} \cdot t} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      5. associate-/l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{4} \cdot t} + \color{blue}{\ell \cdot \frac{\ell}{{k}^{4} \cdot t}} \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t}} \cdot \left(\ell + \ell\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell + \ell\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{{k}^{4}} \cdot t} \cdot \left(\ell + \ell\right) \]

      11. count-2-rev N/A

          \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

      12. lower-*.f64 76.4

          \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 78.3%

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

   if 6.9999999999999998e-162 < l

   1. Initial program 32.0%

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

   3. Taylor expanded in t around 0

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

      1. count-2-rev N/A

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

      2. div-add-rev N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 56.1%

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

   10. Applied rewrites 60.0%

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

Alternative 14: 70.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l (* (* t (* k k)) (* k k))) (* 2.0 l)))

C

double code(double t, double l, double k) {
	return (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double t, double l, double k) {
	return (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
}

Python

def code(t, l, k):
	return (l / ((t * (k * k)) * (k * k))) * (2.0 * l)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(t * Float64(k * k)) * Float64(k * k))) * Float64(2.0 * l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / ((t * (k * k)) * (k * k))) * (2.0 * l);
end

Wolfram

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

Derivation

1. Initial program 35.1%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{{k}^{4} \cdot t}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{4} \cdot t} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{4} \cdot t} + \color{blue}{\ell \cdot \frac{\ell}{{k}^{4} \cdot t}} \]

   6. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t} \cdot \left(\ell + \ell\right)} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{4} \cdot t}} \cdot \left(\ell + \ell\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell + \ell\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{{k}^{4}} \cdot t} \cdot \left(\ell + \ell\right) \]

   11. count-2-rev N/A

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

   12. lower-*.f64 67.7

       \[\leadsto \frac{\ell}{{k}^{4} \cdot t} \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

5. Applied rewrites 67.7%

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

7. Applied rewrites 69.6%

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

Alternative 15: 19.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-0.11666666666666667 \cdot \ell\right) \cdot \ell}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (* (* -0.11666666666666667 l) l) t))

C

double code(double t, double l, double k) {
	return ((-0.11666666666666667 * l) * l) / t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (((-0.11666666666666667d0) * l) * l) / t
end function

Java

public static double code(double t, double l, double k) {
	return ((-0.11666666666666667 * l) * l) / t;
}

Python

def code(t, l, k):
	return ((-0.11666666666666667 * l) * l) / t

Julia

function code(t, l, k)
	return Float64(Float64(Float64(-0.11666666666666667 * l) * l) / t)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((-0.11666666666666667 * l) * l) / t;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(-0.11666666666666667 * l), $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Applied rewrites 32.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k\right) \cdot k, k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 20.3%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
9. Step-by-step derivation

10. Applied rewrites 18.5%

    \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \color{blue}{-0.11666666666666667}\right) \]
11. Step-by-step derivation

12. Applied rewrites 20.3%

    \[\leadsto \frac{\left(-0.11666666666666667 \cdot \ell\right) \cdot \ell}{t} \]
13. Add Preprocessing

Alternative 16: 19.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((-0.11666666666666667d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

def code(t, l, k):
	return (l * l) * (-0.11666666666666667 / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Applied rewrites 32.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k\right) \cdot k, k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 20.3%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
9. Step-by-step derivation

10. Applied rewrites 20.3%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{\color{blue}{t}} \]
11. Add Preprocessing

Alternative 17: 18.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(\ell \cdot \frac{-0.11666666666666667}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* l (/ -0.11666666666666667 t))))

C

double code(double t, double l, double k) {
	return l * (l * (-0.11666666666666667 / t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * (l * ((-0.11666666666666667d0) / t))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l * (-0.11666666666666667 / t));
}

Python

def code(t, l, k):
	return l * (l * (-0.11666666666666667 / t))

Julia

function code(t, l, k)
	return Float64(l * Float64(l * Float64(-0.11666666666666667 / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l * (-0.11666666666666667 / t));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Applied rewrites 32.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k\right) \cdot k, k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 20.3%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
9. Step-by-step derivation

10. Applied rewrites 18.5%

    \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \color{blue}{-0.11666666666666667}\right) \]
11. Step-by-step derivation

12. Applied rewrites 18.5%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{-0.11666666666666667}{\color{blue}{t}}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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))))