Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 93.5%

Time: 7.7s

Alternatives: 9

Speedup: 4.4×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 35.7% accurate, 1.0× speedup

Math

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

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: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\ell}{\left|k\right|}\\ \mathbf{if}\;\left|k\right| \leq 10^{+118}:\\ \;\;\;\;\frac{\ell}{\left|k\right| \cdot \left|k\right|} \cdot \left(\ell \cdot \left(\frac{1}{\tan \left(\left|k\right|\right) \cdot \sin \left(\left|k\right|\right)} \cdot \frac{2}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot \left(\frac{\cos \left(\left|k\right|\right)}{\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t} \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (fabs k))))
   (if (<= (fabs k) 1e+118)
     (*
      (/ l (* (fabs k) (fabs k)))
      (* l (* (/ 1.0 (* (tan (fabs k)) (sin (fabs k)))) (/ 2.0 t))))
     (*
      (* t_1 t_1)
      (*
       (/ (cos (fabs k)) (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t))
       2.0)))))

C

double code(double t, double l, double k) {
	double t_1 = l / fabs(k);
	double tmp;
	if (fabs(k) <= 1e+118) {
		tmp = (l / (fabs(k) * fabs(k))) * (l * ((1.0 / (tan(fabs(k)) * sin(fabs(k)))) * (2.0 / t)));
	} else {
		tmp = (t_1 * t_1) * ((cos(fabs(k)) / ((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t)) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / abs(k)
    if (abs(k) <= 1d+118) then
        tmp = (l / (abs(k) * abs(k))) * (l * ((1.0d0 / (tan(abs(k)) * sin(abs(k)))) * (2.0d0 / t)))
    else
        tmp = (t_1 * t_1) * ((cos(abs(k)) / ((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 1e+118) {
		tmp = (l / (Math.abs(k) * Math.abs(k))) * (l * ((1.0 / (Math.tan(Math.abs(k)) * Math.sin(Math.abs(k)))) * (2.0 / t)));
	} else {
		tmp = (t_1 * t_1) * ((Math.cos(Math.abs(k)) / ((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t)) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 1e+118:
		tmp = (l / (math.fabs(k) * math.fabs(k))) * (l * ((1.0 / (math.tan(math.fabs(k)) * math.sin(math.fabs(k)))) * (2.0 / t)))
	else:
		tmp = (t_1 * t_1) * ((math.cos(math.fabs(k)) / ((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t)) * 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / abs(k))
	tmp = 0.0
	if (abs(k) <= 1e+118)
		tmp = Float64(Float64(l / Float64(abs(k) * abs(k))) * Float64(l * Float64(Float64(1.0 / Float64(tan(abs(k)) * sin(abs(k)))) * Float64(2.0 / t))));
	else
		tmp = Float64(Float64(t_1 * t_1) * Float64(Float64(cos(abs(k)) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / abs(k);
	tmp = 0.0;
	if (abs(k) <= 1e+118)
		tmp = (l / (abs(k) * abs(k))) * (l * ((1.0 / (tan(abs(k)) * sin(abs(k)))) * (2.0 / t)));
	else
		tmp = (t_1 * t_1) * ((cos(abs(k)) / ((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1e+118], N[(N[(l / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(1.0 / N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999997e117

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

      \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.8%

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 86.0%

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

   if 9.9999999999999997e117 < k

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

      \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.6%

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

   8. Applied rewrites 81.6%

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

Alternative 2: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 10^{+118}:\\ \;\;\;\;\frac{\ell}{\left|k\right| \cdot \left|k\right|} \cdot \left(\ell \cdot \left(\frac{1}{\tan \left(\left|k\right|\right) \cdot \sin \left(\left|k\right|\right)} \cdot \frac{2}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\ell}{\left|k\right|}}{\left|k\right|}\right) \cdot \left(\frac{\cos \left(\left|k\right|\right)}{\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t} \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1e+118)
   (*
    (/ l (* (fabs k) (fabs k)))
    (* l (* (/ 1.0 (* (tan (fabs k)) (sin (fabs k)))) (/ 2.0 t))))
   (*
    (* l (/ (/ l (fabs k)) (fabs k)))
    (*
     (/ (cos (fabs k)) (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t))
     2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1e+118) {
		tmp = (l / (fabs(k) * fabs(k))) * (l * ((1.0 / (tan(fabs(k)) * sin(fabs(k)))) * (2.0 / t)));
	} else {
		tmp = (l * ((l / fabs(k)) / fabs(k))) * ((cos(fabs(k)) / ((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t)) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 1d+118) then
        tmp = (l / (abs(k) * abs(k))) * (l * ((1.0d0 / (tan(abs(k)) * sin(abs(k)))) * (2.0d0 / t)))
    else
        tmp = (l * ((l / abs(k)) / abs(k))) * ((cos(abs(k)) / ((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 1e+118) {
		tmp = (l / (Math.abs(k) * Math.abs(k))) * (l * ((1.0 / (Math.tan(Math.abs(k)) * Math.sin(Math.abs(k)))) * (2.0 / t)));
	} else {
		tmp = (l * ((l / Math.abs(k)) / Math.abs(k))) * ((Math.cos(Math.abs(k)) / ((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t)) * 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 1e+118:
		tmp = (l / (math.fabs(k) * math.fabs(k))) * (l * ((1.0 / (math.tan(math.fabs(k)) * math.sin(math.fabs(k)))) * (2.0 / t)))
	else:
		tmp = (l * ((l / math.fabs(k)) / math.fabs(k))) * ((math.cos(math.fabs(k)) / ((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t)) * 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1e+118)
		tmp = Float64(Float64(l / Float64(abs(k) * abs(k))) * Float64(l * Float64(Float64(1.0 / Float64(tan(abs(k)) * sin(abs(k)))) * Float64(2.0 / t))));
	else
		tmp = Float64(Float64(l * Float64(Float64(l / abs(k)) / abs(k))) * Float64(Float64(cos(abs(k)) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 1e+118)
		tmp = (l / (abs(k) * abs(k))) * (l * ((1.0 / (tan(abs(k)) * sin(abs(k)))) * (2.0 / t)));
	else
		tmp = (l * ((l / abs(k)) / abs(k))) * ((cos(abs(k)) / ((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1e+118], N[(N[(l / N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(1.0 / N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(l / N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999997e117

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

      \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 75.8%

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 86.0%

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

   if 9.9999999999999997e117 < k

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

      \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.1%

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

   8. Applied rewrites 80.1%

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

Alternative 3: 86.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (l / (k * k)) * (l * ((1.0 / (tan(k) * sin(k))) * (2.0 / 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 / (k * k)) * (l * ((1.0d0 / (tan(k) * sin(k))) * (2.0d0 / t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 73.3%

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

4. Applied rewrites 73.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 74.3%

   \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 75.8%

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lift-*.f64 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 86.0%

   \[\leadsto \frac{\ell}{k \cdot k} \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{\tan k \cdot \sin k} \cdot \frac{2}{t}\right)\right)} \]
9. Add Preprocessing

Alternative 4: 83.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := {\left(\left|k\right|\right)}^{2}\\ t_2 := \left|k\right| \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\left(\ell \cdot \frac{\ell}{t\_2}\right) \cdot \left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{t\_1}{t}, \frac{1}{t}\right)}{t\_1} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\cos \left(\left|k\right|\right) \cdot 2\right)}{\left(t\_2 \cdot t\right) \cdot \mathsf{fma}\left(-0.5, \cos \left(\left|k\right| + \left|k\right|\right), 0.5\right)} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (fabs k) 2.0)) (t_2 (* (fabs k) (fabs k))))
   (if (<= (fabs k) 3.7e-7)
     (*
      (* l (/ l t_2))
      (* (/ (fma -0.16666666666666666 (/ t_1 t) (/ 1.0 t)) t_1) 2.0))
     (*
      (/
       (* l (* (cos (fabs k)) 2.0))
       (* (* t_2 t) (fma -0.5 (cos (+ (fabs k) (fabs k))) 0.5)))
      l))))

C

double code(double t, double l, double k) {
	double t_1 = pow(fabs(k), 2.0);
	double t_2 = fabs(k) * fabs(k);
	double tmp;
	if (fabs(k) <= 3.7e-7) {
		tmp = (l * (l / t_2)) * ((fma(-0.16666666666666666, (t_1 / t), (1.0 / t)) / t_1) * 2.0);
	} else {
		tmp = ((l * (cos(fabs(k)) * 2.0)) / ((t_2 * t) * fma(-0.5, cos((fabs(k) + fabs(k))), 0.5))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = abs(k) ^ 2.0
	t_2 = Float64(abs(k) * abs(k))
	tmp = 0.0
	if (abs(k) <= 3.7e-7)
		tmp = Float64(Float64(l * Float64(l / t_2)) * Float64(Float64(fma(-0.16666666666666666, Float64(t_1 / t), Float64(1.0 / t)) / t_1) * 2.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(cos(abs(k)) * 2.0)) / Float64(Float64(t_2 * t) * fma(-0.5, cos(Float64(abs(k) + abs(k))), 0.5))) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 3.7e-7], N[(N[(l * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * N[(t$95$1 / t), $MachinePrecision] + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t), $MachinePrecision] * N[(-0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-7

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 55.5%

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

   9. Applied rewrites 55.5%

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

   if 3.7e-7 < k

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

   8. Applied rewrites 74.5%

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

Alternative 5: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.7 \cdot 10^{-165}:\\ \;\;\;\;\left(\left|\ell\right| \cdot \frac{\left|\ell\right|}{k \cdot k}\right) \cdot \frac{2}{{k}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot \frac{1}{\tan k \cdot \sin k}}{\left(k \cdot k\right) \cdot t} \cdot 2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 3.7e-165)
   (* (* (fabs l) (/ (fabs l) (* k k))) (/ 2.0 (* (pow k 2.0) t)))
   (*
    (/ (* (* (fabs l) (fabs l)) (/ 1.0 (* (tan k) (sin k)))) (* (* k k) t))
    2.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 3.7e-165) {
		tmp = (fabs(l) * (fabs(l) / (k * k))) * (2.0 / (pow(k, 2.0) * t));
	} else {
		tmp = (((fabs(l) * fabs(l)) * (1.0 / (tan(k) * sin(k)))) / ((k * k) * t)) * 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(l) <= 3.7d-165) then
        tmp = (abs(l) * (abs(l) / (k * k))) * (2.0d0 / ((k ** 2.0d0) * t))
    else
        tmp = (((abs(l) * abs(l)) * (1.0d0 / (tan(k) * sin(k)))) / ((k * k) * t)) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 3.7e-165) {
		tmp = (Math.abs(l) * (Math.abs(l) / (k * k))) * (2.0 / (Math.pow(k, 2.0) * t));
	} else {
		tmp = (((Math.abs(l) * Math.abs(l)) * (1.0 / (Math.tan(k) * Math.sin(k)))) / ((k * k) * t)) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 3.7e-165:
		tmp = (math.fabs(l) * (math.fabs(l) / (k * k))) * (2.0 / (math.pow(k, 2.0) * t))
	else:
		tmp = (((math.fabs(l) * math.fabs(l)) * (1.0 / (math.tan(k) * math.sin(k)))) / ((k * k) * t)) * 2.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 3.7e-165)
		tmp = Float64(Float64(abs(l) * Float64(abs(l) / Float64(k * k))) * Float64(2.0 / Float64((k ^ 2.0) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(abs(l) * abs(l)) * Float64(1.0 / Float64(tan(k) * sin(k)))) / Float64(Float64(k * k) * t)) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 3.7e-165)
		tmp = (abs(l) * (abs(l) / (k * k))) * (2.0 / ((k ^ 2.0) * t));
	else
		tmp = (((abs(l) * abs(l)) * (1.0 / (tan(k) * sin(k)))) / ((k * k) * t)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 3.7e-165], N[(N[(N[Abs[l], $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.7e-165

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 72.0%

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

   9. Applied rewrites 72.0%

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

   if 3.7e-165 < l

   1. Initial program 35.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.3%

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 74.3%

      \[\leadsto \left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 74.2%

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

Alternative 6: 72.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 73.3%

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

4. Applied rewrites 73.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 74.3%

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

7. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 72.0%

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

9. Applied rewrites 72.0%

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

Alternative 7: 69.6% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (+ l l) (/ (* (pow k -4.0) l) t)))

C

double code(double t, double l, double k) {
	return (l + l) * ((pow(k, -4.0) * 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 = (l + l) * (((k ** (-4.0d0)) * l) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * ((Math.pow(k, -4.0) * l) / t);
}

Python

def code(t, l, k):
	return (l + l) * ((math.pow(k, -4.0) * l) / t)

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(Float64((k ^ -4.0) * l) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (((k ^ -4.0) * l) / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(N[(N[Power[k, -4.0], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 62.7%

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

4. Applied rewrites 62.7%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-flip N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 68.8%

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

6. Applied rewrites 68.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 68.8%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 68.8%

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

8. Applied rewrites 68.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 69.6%

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

10. Applied rewrites 69.6%

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

Alternative 8: 68.9% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (+ l l) (* (/ (pow k -4.0) t) l)))

C

double code(double t, double l, double k) {
	return (l + l) * ((pow(k, -4.0) / t) * 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 + l) * (((k ** (-4.0d0)) / t) * l)
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * ((Math.pow(k, -4.0) / t) * l);
}

Python

def code(t, l, k):
	return (l + l) * ((math.pow(k, -4.0) / t) * l)

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(Float64((k ^ -4.0) / t) * l))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (((k ^ -4.0) / t) * l);
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(N[(N[Power[k, -4.0], $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 62.7%

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

4. Applied rewrites 62.7%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-flip N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 68.8%

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

6. Applied rewrites 68.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 68.8%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 68.8%

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

8. Applied rewrites 68.8%

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

Alternative 9: 68.8% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (+ l l) (/ l (* (pow k 4.0) t))))

C

double code(double t, double l, double k) {
	return (l + l) * (l / (pow(k, 4.0) * 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) * (l / ((k ** 4.0d0) * t))
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * (l / (Math.pow(k, 4.0) * t));
}

Python

def code(t, l, k):
	return (l + l) * (l / (math.pow(k, 4.0) * t))

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(l / Float64((k ^ 4.0) * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (l / ((k ^ 4.0) * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 62.7%

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

4. Applied rewrites 62.7%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. mult-flip N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-flip N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 68.8%

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

6. Applied rewrites 68.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 68.8%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 68.8%

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

8. Applied rewrites 68.8%

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

9. Taylor expanded in t around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 68.9%

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

11. Applied rewrites 68.9%

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

Reproduce

herbie shell --seed 2025196 
(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))))