Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 89.4%

Time: 7.8s

Alternatives: 11

Speedup: 4.4×

• 39.1% 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: 35.2% 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: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 2 \cdot 10^{+67}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot t\_1}{\left(\left|k\right| \cdot \left|k\right|\right) \cdot \left(t \cdot {\sin \left(\left|k\right|\right)}^{2}\right)}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(t\_1 \cdot \ell\right) \cdot 2}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|}}{\left|k\right|} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cos (fabs k))))
   (if (<= (fabs k) 2e+67)
     (*
      (*
       2.0
       (/ (* l t_1) (* (* (fabs k) (fabs k)) (* t (pow (sin (fabs k)) 2.0)))))
      l)
     (*
      (/
       (/
        (* (* t_1 l) 2.0)
        (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k)))
       (fabs k))
      l))))

C

double code(double t, double l, double k) {
	double t_1 = cos(fabs(k));
	double tmp;
	if (fabs(k) <= 2e+67) {
		tmp = (2.0 * ((l * t_1) / ((fabs(k) * fabs(k)) * (t * pow(sin(fabs(k)), 2.0))))) * l;
	} else {
		tmp = ((((t_1 * l) * 2.0) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k))) / fabs(k)) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cos(abs(k))
	tmp = 0.0
	if (abs(k) <= 2e+67)
		tmp = Float64(Float64(2.0 * Float64(Float64(l * t_1) / Float64(Float64(abs(k) * abs(k)) * Float64(t * (sin(abs(k)) ^ 2.0))))) * l);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 * l) * 2.0) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))) / abs(k)) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 2e+67], N[(N[(2.0 * N[(N[(l * t$95$1), $MachinePrecision] / N[(N[(N[Abs[k], $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[Power[N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(N[(N[(t$95$1 * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999997e67

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

   if 1.99999999999999997e67 < k

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 82.3

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

   8. Applied rewrites 82.3%

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

   10. Applied rewrites 82.2%

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

Alternative 2: 89.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot 1}{{\left(\left|k\right|\right)}^{2} \cdot \left(t \cdot {\sin \left(\left|k\right|\right)}^{2}\right)}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos \left(\left|k\right|\right) \cdot \ell\right) \cdot 2}{\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|}}{\left|k\right|} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 1.4e-8)
   (*
    (* 2.0 (/ (* l 1.0) (* (pow (fabs k) 2.0) (* t (pow (sin (fabs k)) 2.0)))))
    l)
   (*
    (/
     (/
      (* (* (cos (fabs k)) l) 2.0)
      (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k)))
     (fabs k))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 1.4e-8) {
		tmp = (2.0 * ((l * 1.0) / (pow(fabs(k), 2.0) * (t * pow(sin(fabs(k)), 2.0))))) * l;
	} else {
		tmp = ((((cos(fabs(k)) * l) * 2.0) / ((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k))) / fabs(k)) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 1.4e-8)
		tmp = Float64(Float64(2.0 * Float64(Float64(l * 1.0) / Float64((abs(k) ^ 2.0) * Float64(t * (sin(abs(k)) ^ 2.0))))) * l);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(abs(k)) * l) * 2.0) / Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k))) / abs(k)) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 1.4e-8], N[(N[(2.0 * N[(N[(l * 1.0), $MachinePrecision] / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[k], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-8

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 72.0%

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

   if 1.4e-8 < k

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 82.3

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

   8. Applied rewrites 82.3%

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

   10. Applied rewrites 82.2%

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

Alternative 3: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot 1}{{\left(\left|k\right|\right)}^{2} \cdot \left(t \cdot {\sin \left(\left|k\right|\right)}^{2}\right)}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\left|k\right|\right) \cdot \left(\ell \cdot \frac{2}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|}\right)\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 3.7e-8)
   (*
    (* 2.0 (/ (* l 1.0) (* (pow (fabs k) 2.0) (* t (pow (sin (fabs k)) 2.0)))))
    l)
   (*
    (*
     (cos (fabs k))
     (*
      l
      (/
       2.0
       (*
        (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k))
        (fabs k)))))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 3.7e-8) {
		tmp = (2.0 * ((l * 1.0) / (pow(fabs(k), 2.0) * (t * pow(sin(fabs(k)), 2.0))))) * l;
	} else {
		tmp = (cos(fabs(k)) * (l * (2.0 / (((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k)) * fabs(k))))) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 3.7e-8)
		tmp = Float64(Float64(2.0 * Float64(Float64(l * 1.0) / Float64((abs(k) ^ 2.0) * Float64(t * (sin(abs(k)) ^ 2.0))))) * l);
	else
		tmp = Float64(Float64(cos(abs(k)) * Float64(l * Float64(2.0 / Float64(Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k)) * abs(k))))) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 3.7e-8], N[(N[(2.0 * N[(N[(l * 1.0), $MachinePrecision] / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * N[(l * N[(2.0 / N[(N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-8

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 72.0%

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

   if 3.7e-8 < k

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 82.3

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

   8. Applied rewrites 82.3%

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

   10. Applied rewrites 78.3%

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

Alternative 4: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\left(2 \cdot \frac{\ell \cdot 1}{{\left(\left|k\right|\right)}^{2} \cdot \left(t \cdot {\sin \left(\left|k\right|\right)}^{2}\right)}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos \left(\left|k\right|\right)}{\left(\left(\mathsf{fma}\left(\cos \left(\left|k\right| + \left|k\right|\right), -0.5, 0.5\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|} \cdot \left(\ell + \ell\right)\right) \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 3.7e-8)
   (*
    (* 2.0 (/ (* l 1.0) (* (pow (fabs k) 2.0) (* t (pow (sin (fabs k)) 2.0)))))
    l)
   (*
    (*
     (/
      (cos (fabs k))
      (*
       (* (* (fma (cos (+ (fabs k) (fabs k))) -0.5 0.5) t) (fabs k))
       (fabs k)))
     (+ l l))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 3.7e-8) {
		tmp = (2.0 * ((l * 1.0) / (pow(fabs(k), 2.0) * (t * pow(sin(fabs(k)), 2.0))))) * l;
	} else {
		tmp = ((cos(fabs(k)) / (((fma(cos((fabs(k) + fabs(k))), -0.5, 0.5) * t) * fabs(k)) * fabs(k))) * (l + l)) * l;
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 3.7e-8)
		tmp = Float64(Float64(2.0 * Float64(Float64(l * 1.0) / Float64((abs(k) ^ 2.0) * Float64(t * (sin(abs(k)) ^ 2.0))))) * l);
	else
		tmp = Float64(Float64(Float64(cos(abs(k)) / Float64(Float64(Float64(fma(cos(Float64(abs(k) + abs(k))), -0.5, 0.5) * t) * abs(k)) * abs(k))) * Float64(l + l)) * l);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 3.7e-8], N[(N[(2.0 * N[(N[(l * 1.0), $MachinePrecision] / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-8

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 72.0%

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

   if 3.7e-8 < k

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 82.3

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

   8. Applied rewrites 82.3%

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

   10. Applied rewrites 78.3%

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

Alternative 5: 82.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sin (fabs k))))
   (if (<= (fabs k) 2e-8)
     (* (* 2.0 (/ (* l 1.0) (* (pow (fabs k) 2.0) (* t (pow t_1 2.0))))) l)
     (/
      2.0
      (* (fabs k) (* (fabs k) (* (/ t (* l l)) (* (tan (fabs k)) t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 2e-8) {
		tmp = (2.0 * ((l * 1.0) / (pow(fabs(k), 2.0) * (t * pow(t_1, 2.0))))) * l;
	} else {
		tmp = 2.0 / (fabs(k) * (fabs(k) * ((t / (l * l)) * (tan(fabs(k)) * t_1))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 2e-8) {
		tmp = (2.0 * ((l * 1.0) / (Math.pow(Math.abs(k), 2.0) * (t * Math.pow(t_1, 2.0))))) * l;
	} else {
		tmp = 2.0 / (Math.abs(k) * (Math.abs(k) * ((t / (l * l)) * (Math.tan(Math.abs(k)) * t_1))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 2e-8:
		tmp = (2.0 * ((l * 1.0) / (math.pow(math.fabs(k), 2.0) * (t * math.pow(t_1, 2.0))))) * l
	else:
		tmp = 2.0 / (math.fabs(k) * (math.fabs(k) * ((t / (l * l)) * (math.tan(math.fabs(k)) * t_1))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 2e-8)
		tmp = Float64(Float64(2.0 * Float64(Float64(l * 1.0) / Float64((abs(k) ^ 2.0) * Float64(t * (t_1 ^ 2.0))))) * l);
	else
		tmp = Float64(2.0 / Float64(abs(k) * Float64(abs(k) * Float64(Float64(t / Float64(l * l)) * Float64(tan(abs(k)) * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 2e-8)
		tmp = (2.0 * ((l * 1.0) / ((abs(k) ^ 2.0) * (t * (t_1 ^ 2.0))))) * l;
	else
		tmp = 2.0 / (abs(k) * (abs(k) * ((t / (l * l)) * (tan(abs(k)) * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 2e-8], N[(N[(2.0 * N[(N[(l * 1.0), $MachinePrecision] / N[(N[Power[N[Abs[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2e-8

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 72.0%

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

   if 2e-8 < k

   1. Initial program 35.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. tan-quot N/A

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

      18. lift-tan.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 74.8%

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

Alternative 6: 72.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 45:\\ \;\;\;\;\frac{\left(\cos \left(\left|k\right|\right) \cdot \ell\right) \cdot 2}{\left({\left(\left|k\right|\right)}^{3} \cdot t\right) \cdot \left|k\right|} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 45.0)
   (* (/ (* (* (cos (fabs k)) l) 2.0) (* (* (pow (fabs k) 3.0) t) (fabs k))) l)
   (*
    (/
     (* 2.0 l)
     (*
      (* (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t) (fabs k))
      (fabs k)))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 45.0) {
		tmp = (((cos(fabs(k)) * l) * 2.0) / ((pow(fabs(k), 3.0) * t) * fabs(k))) * l;
	} else {
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t) * fabs(k)) * fabs(k))) * l;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 45.0d0) then
        tmp = (((cos(abs(k)) * l) * 2.0d0) / (((abs(k) ** 3.0d0) * t) * abs(k))) * l
    else
        tmp = ((2.0d0 * l) / ((((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t) * abs(k)) * abs(k))) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 45.0) {
		tmp = (((Math.cos(Math.abs(k)) * l) * 2.0) / ((Math.pow(Math.abs(k), 3.0) * t) * Math.abs(k))) * l;
	} else {
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t) * Math.abs(k)) * Math.abs(k))) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 45.0:
		tmp = (((math.cos(math.fabs(k)) * l) * 2.0) / ((math.pow(math.fabs(k), 3.0) * t) * math.fabs(k))) * l
	else:
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t) * math.fabs(k)) * math.fabs(k))) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 45.0)
		tmp = Float64(Float64(Float64(Float64(cos(abs(k)) * l) * 2.0) / Float64(Float64((abs(k) ^ 3.0) * t) * abs(k))) * l);
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t) * abs(k)) * abs(k))) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 45.0)
		tmp = (((cos(abs(k)) * l) * 2.0) / (((abs(k) ^ 3.0) * t) * abs(k))) * l;
	else
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t) * abs(k)) * abs(k))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 45.0], N[(N[(N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Power[N[Abs[k], $MachinePrecision], 3.0], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 45

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.3

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot 2}{\left({k}^{3} \cdot t\right) \cdot k} \cdot \ell \]

       2. lower-pow.f64 71.0

          \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot 2}{\left({k}^{3} \cdot t\right) \cdot k} \cdot \ell \]

   11. Applied rewrites 71.0%

       \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot 2}{\left({k}^{3} \cdot t\right) \cdot k} \cdot \ell \]

   if 45 < k

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.3

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 65.2

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

   11. Applied rewrites 65.2%

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

Alternative 7: 72.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (2.0 * ((l * 1.0) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))))) * l;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 * ((l * 1.0d0) / ((k ** 2.0d0) * (t * (sin(k) ** 2.0d0))))) * l
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. associate-*l/ N/A

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

   12. associate-/r/ N/A

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

3. Applied rewrites 38.0%

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

4. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-sin.f64 82.3

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

6. Applied rewrites 82.3%

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

7. Taylor expanded in k around 0

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

9. Applied rewrites 72.0%

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

Alternative 8: 71.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 250:\\ \;\;\;\;\left(\frac{{\left(\left|k\right|\right)}^{-4} \cdot \ell}{t} \cdot \ell\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(\left|k\right| + \left|k\right|\right)\right) \cdot t\right) \cdot \left|k\right|\right) \cdot \left|k\right|} \cdot \ell\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs k) 250.0)
   (* (* (/ (* (pow (fabs k) -4.0) l) t) l) 2.0)
   (*
    (/
     (* 2.0 l)
     (*
      (* (* (- 0.5 (* 0.5 (cos (+ (fabs k) (fabs k))))) t) (fabs k))
      (fabs k)))
    l)))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 250.0) {
		tmp = (((pow(fabs(k), -4.0) * l) / t) * l) * 2.0;
	} else {
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * cos((fabs(k) + fabs(k))))) * t) * fabs(k)) * fabs(k))) * l;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(k) <= 250.0d0) then
        tmp = ((((abs(k) ** (-4.0d0)) * l) / t) * l) * 2.0d0
    else
        tmp = ((2.0d0 * l) / ((((0.5d0 - (0.5d0 * cos((abs(k) + abs(k))))) * t) * abs(k)) * abs(k))) * l
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(k) <= 250.0) {
		tmp = (((Math.pow(Math.abs(k), -4.0) * l) / t) * l) * 2.0;
	} else {
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * Math.cos((Math.abs(k) + Math.abs(k))))) * t) * Math.abs(k)) * Math.abs(k))) * l;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(k) <= 250.0:
		tmp = (((math.pow(math.fabs(k), -4.0) * l) / t) * l) * 2.0
	else:
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * math.cos((math.fabs(k) + math.fabs(k))))) * t) * math.fabs(k)) * math.fabs(k))) * l
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 250.0)
		tmp = Float64(Float64(Float64(Float64((abs(k) ^ -4.0) * l) / t) * l) * 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(abs(k) + abs(k))))) * t) * abs(k)) * abs(k))) * l);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(k) <= 250.0)
		tmp = ((((abs(k) ^ -4.0) * l) / t) * l) * 2.0;
	else
		tmp = ((2.0 * l) / ((((0.5 - (0.5 * cos((abs(k) + abs(k))))) * t) * abs(k)) * abs(k))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 250.0], N[(N[(N[(N[(N[Power[N[Abs[k], $MachinePrecision], -4.0], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 250

   1. Initial program 35.2%

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

   2. Taylor expanded in 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-rgt-out N/A

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

      8. count-2-rev N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.6

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 62.6

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

   6. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 69.3

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

   10. Applied rewrites 69.3%

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

   if 250 < k

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.3

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 65.2

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

   11. Applied rewrites 65.2%

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

Alternative 9: 71.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 2.6 \cdot 10^{+172}:\\ \;\;\;\;\left(\frac{{k}^{-4} \cdot \left|\ell\right|}{t} \cdot \left|\ell\right|\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k \cdot \left|\ell\right|\right) \cdot 2}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\right) \cdot k} \cdot \left|\ell\right|\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (fabs l) 2.6e+172)
   (* (* (/ (* (pow k -4.0) (fabs l)) t) (fabs l)) 2.0)
   (*
    (/ (* (* (cos k) (fabs l)) 2.0) (* (* (* (- 0.5 0.5) t) k) k))
    (fabs l))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(l) <= 2.6e+172) {
		tmp = (((pow(k, -4.0) * fabs(l)) / t) * fabs(l)) * 2.0;
	} else {
		tmp = (((cos(k) * fabs(l)) * 2.0) / ((((0.5 - 0.5) * t) * k) * k)) * fabs(l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (abs(l) <= 2.6d+172) then
        tmp = ((((k ** (-4.0d0)) * abs(l)) / t) * abs(l)) * 2.0d0
    else
        tmp = (((cos(k) * abs(l)) * 2.0d0) / ((((0.5d0 - 0.5d0) * t) * k) * k)) * abs(l)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(l) <= 2.6e+172) {
		tmp = (((Math.pow(k, -4.0) * Math.abs(l)) / t) * Math.abs(l)) * 2.0;
	} else {
		tmp = (((Math.cos(k) * Math.abs(l)) * 2.0) / ((((0.5 - 0.5) * t) * k) * k)) * Math.abs(l);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(l) <= 2.6e+172:
		tmp = (((math.pow(k, -4.0) * math.fabs(l)) / t) * math.fabs(l)) * 2.0
	else:
		tmp = (((math.cos(k) * math.fabs(l)) * 2.0) / ((((0.5 - 0.5) * t) * k) * k)) * math.fabs(l)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(l) <= 2.6e+172)
		tmp = Float64(Float64(Float64(Float64((k ^ -4.0) * abs(l)) / t) * abs(l)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * abs(l)) * 2.0) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k) * k)) * abs(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(l) <= 2.6e+172)
		tmp = ((((k ^ -4.0) * abs(l)) / t) * abs(l)) * 2.0;
	else
		tmp = (((cos(k) * abs(l)) * 2.0) / ((((0.5 - 0.5) * t) * k) * k)) * abs(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[l], $MachinePrecision], 2.6e+172], N[(N[(N[(N[(N[Power[k, -4.0], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.6e172

   1. Initial program 35.2%

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

   2. Taylor expanded in 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

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-rgt-out N/A

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

      8. count-2-rev N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-flip N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 62.6

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 62.6

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

   6. Applied rewrites 62.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 68.6

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

   8. Applied rewrites 68.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 69.3

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

   10. Applied rewrites 69.3%

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

   if 2.6e172 < l

   1. Initial program 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. associate-/r/ N/A

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 82.3

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

   6. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 82.3

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 40.8%

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

Alternative 10: 69.3% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (* (/ (* (pow k -4.0) l) t) l) 2.0))

C

double code(double t, double l, double k) {
	return (((pow(k, -4.0) * l) / t) * l) * 2.0;
}

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 = ((((k ** (-4.0d0)) * l) / t) * l) * 2.0d0
end function

Java

public static double code(double t, double l, double k) {
	return (((Math.pow(k, -4.0) * l) / t) * l) * 2.0;
}

Python

def code(t, l, k):
	return (((math.pow(k, -4.0) * l) / t) * l) * 2.0

Julia

function code(t, l, k)
	return Float64(Float64(Float64(Float64((k ^ -4.0) * l) / t) * l) * 2.0)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((((k ^ -4.0) * l) / t) * l) * 2.0;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[(N[Power[k, -4.0], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

2. Taylor expanded in 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

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

   2. count-2-rev N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. distribute-rgt-out N/A

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

   8. count-2-rev N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-flip N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 62.6

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. lift-*.f64 62.6

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

6. Applied rewrites 62.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 68.6

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

8. Applied rewrites 68.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 69.3

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

10. Applied rewrites 69.3%

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

Alternative 11: 68.8% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (* (/ l (* (pow k 4.0) t)) l) 2.0))

C

double code(double t, double l, double k) {
	return ((l / (pow(k, 4.0) * t)) * l) * 2.0;
}

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 = ((l / ((k ** 4.0d0) * t)) * l) * 2.0d0
end function

Java

public static double code(double t, double l, double k) {
	return ((l / (Math.pow(k, 4.0) * t)) * l) * 2.0;
}

Python

def code(t, l, k):
	return ((l / (math.pow(k, 4.0) * t)) * l) * 2.0

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / Float64((k ^ 4.0) * t)) * l) * 2.0)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / ((k ^ 4.0) * t)) * l) * 2.0;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / N[(N[Power[k, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

2. Taylor expanded in 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

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

   2. count-2-rev N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. distribute-rgt-out N/A

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

   8. count-2-rev N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-flip N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 62.6

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. lift-*.f64 62.6

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

6. Applied rewrites 62.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 68.6

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

8. Applied rewrites 68.6%

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

9. Taylor expanded in t around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 68.8

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

11. Applied rewrites 68.8%

    \[\leadsto \left(\frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right) \cdot 2 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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))))