Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 91.0%

Time: 7.1s

Alternatives: 18

Speedup: 12.5×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.0% accurate, 0.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e+157)
   (/
    2.0
    (*
     (/
      (fma (pow (* (sin k_m) t) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
      (* (cos k_m) l))
     (/ t l)))
   (*
    (/ (* (* (/ l k_m) (/ l k_m)) (cos k_m)) (* (pow (sin k_m) 2.0) t))
    2.0)))

C

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

Julia

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e+157], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.20000000000000015e157

   1. Initial program 57.9%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

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

   6. Applied rewrites 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 90.8%

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

   if 9.20000000000000015e157 < k

   1. Initial program 46.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cos.f64 N/A

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

   6. Applied rewrites 91.6%

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

Alternative 2: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\\ \mathbf{if}\;k\_m \leq 47000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{elif}\;k\_m \leq 4.6 \cdot 10^{+160}:\\ \;\;\;\;\left(\left(\ell \cdot \frac{\ell}{k\_m \cdot k\_m}\right) \cdot t\_1\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell \cdot \ell}{k\_m}}{k\_m} \cdot t\_1\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (cos k_m) (* (pow (sin k_m) 2.0) t))))
   (if (<= k_m 47000000000000.0)
     (/ 2.0 (* (* (* (* t (/ t l)) (* (/ t l) (sin k_m))) (tan k_m)) 2.0))
     (if (<= k_m 4.6e+160)
       (* (* (* l (/ l (* k_m k_m))) t_1) 2.0)
       (* (* (/ (/ (* l l) k_m) k_m) t_1) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) / (pow(sin(k_m), 2.0) * t);
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	} else if (k_m <= 4.6e+160) {
		tmp = ((l * (l / (k_m * k_m))) * t_1) * 2.0;
	} else {
		tmp = ((((l * l) / k_m) / k_m) * t_1) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) / ((sin(k_m) ** 2.0d0) * t)
    if (k_m <= 47000000000000.0d0) then
        tmp = 2.0d0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0d0)
    else if (k_m <= 4.6d+160) then
        tmp = ((l * (l / (k_m * k_m))) * t_1) * 2.0d0
    else
        tmp = ((((l * l) / k_m) / k_m) * t_1) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t);
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * Math.sin(k_m))) * Math.tan(k_m)) * 2.0);
	} else if (k_m <= 4.6e+160) {
		tmp = ((l * (l / (k_m * k_m))) * t_1) * 2.0;
	} else {
		tmp = ((((l * l) / k_m) / k_m) * t_1) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t)
	tmp = 0
	if k_m <= 47000000000000.0:
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * math.sin(k_m))) * math.tan(k_m)) * 2.0)
	elif k_m <= 4.6e+160:
		tmp = ((l * (l / (k_m * k_m))) * t_1) * 2.0
	else:
		tmp = ((((l * l) / k_m) / k_m) * t_1) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))
	tmp = 0.0
	if (k_m <= 47000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k_m))) * tan(k_m)) * 2.0));
	elseif (k_m <= 4.6e+160)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k_m * k_m))) * t_1) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / k_m) / k_m) * t_1) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) / ((sin(k_m) ^ 2.0) * t);
	tmp = 0.0;
	if (k_m <= 47000000000000.0)
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	elseif (k_m <= 4.6e+160)
		tmp = ((l * (l / (k_m * k_m))) * t_1) * 2.0;
	else
		tmp = ((((l * l) / k_m) / k_m) * t_1) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 47000000000000.0], N[(2.0 / N[(N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.6e+160], N[(N[(N[(l * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.7e13

   1. Initial program 63.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 4.7e13 < k < 4.59999999999999975e160

   1. Initial program 45.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 76.0

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

   6. Applied rewrites 76.0%

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

   if 4.59999999999999975e160 < k

   1. Initial program 47.1%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

Alternative 3: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 47000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \cos k\_m}{{\sin k\_m}^{2} \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 47000000000000.0)
   (/ 2.0 (* (* (* (* t (/ t l)) (* (/ t l) (sin k_m))) (tan k_m)) 2.0))
   (*
    (/ (* (* (/ l k_m) (/ l k_m)) (cos k_m)) (* (pow (sin k_m) 2.0) t))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	} else {
		tmp = ((((l / k_m) * (l / k_m)) * cos(k_m)) / (pow(sin(k_m), 2.0) * t)) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 47000000000000.0d0) then
        tmp = 2.0d0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0d0)
    else
        tmp = ((((l / k_m) * (l / k_m)) * cos(k_m)) / ((sin(k_m) ** 2.0d0) * t)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * Math.sin(k_m))) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = ((((l / k_m) * (l / k_m)) * Math.cos(k_m)) / (Math.pow(Math.sin(k_m), 2.0) * t)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 47000000000000.0:
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * math.sin(k_m))) * math.tan(k_m)) * 2.0)
	else:
		tmp = ((((l / k_m) * (l / k_m)) * math.cos(k_m)) / (math.pow(math.sin(k_m), 2.0) * t)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 47000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k_m))) * tan(k_m)) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * cos(k_m)) / Float64((sin(k_m) ^ 2.0) * t)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 47000000000000.0)
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	else
		tmp = ((((l / k_m) * (l / k_m)) * cos(k_m)) / ((sin(k_m) ^ 2.0) * t)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 47000000000000.0], N[(2.0 / N[(N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7e13

   1. Initial program 63.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 4.7e13 < k

   1. Initial program 46.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cos.f64 N/A

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

   6. Applied rewrites 84.9%

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

Alternative 4: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 47000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\ell \cdot \frac{\ell}{k\_m \cdot k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 47000000000000.0)
   (/ 2.0 (* (* (* (* t (/ t l)) (* (/ t l) (sin k_m))) (tan k_m)) 2.0))
   (*
    (* (* l (/ l (* k_m k_m))) (/ (cos k_m) (* (pow (sin k_m) 2.0) t)))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	} else {
		tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 47000000000000.0d0) then
        tmp = 2.0d0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0d0)
    else
        tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * Math.sin(k_m))) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = ((l * (l / (k_m * k_m))) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 47000000000000.0:
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * math.sin(k_m))) * math.tan(k_m)) * 2.0)
	else:
		tmp = ((l * (l / (k_m * k_m))) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 47000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k_m))) * tan(k_m)) * 2.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k_m * k_m))) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 47000000000000.0)
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	else
		tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 47000000000000.0], N[(2.0 / N[(N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7e13

   1. Initial program 63.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 4.7e13 < k

   1. Initial program 46.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 70.7

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

   6. Applied rewrites 70.7%

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

Alternative 5: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 47000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot {\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 47000000000000.0)
   (/ 2.0 (* (* (* (* t (/ t l)) (* (/ t l) (sin k_m))) (tan k_m)) 2.0))
   (/
    (* 2.0 (* (* (cos k_m) l) l))
    (* (* (* k_m k_m) t) (pow (sin k_m) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * pow(sin(k_m), 2.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 47000000000000.0d0) then
        tmp = 2.0d0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0d0)
    else
        tmp = (2.0d0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (sin(k_m) ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * Math.sin(k_m))) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * Math.pow(Math.sin(k_m), 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 47000000000000.0:
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * math.sin(k_m))) * math.tan(k_m)) * 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * math.pow(math.sin(k_m), 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 47000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k_m))) * tan(k_m)) * 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * (sin(k_m) ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 47000000000000.0)
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	else
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (sin(k_m) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 47000000000000.0], N[(2.0 / N[(N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7e13

   1. Initial program 63.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 4.7e13 < k

   1. Initial program 46.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   6. Applied rewrites 69.2%

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

Alternative 6: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 47000000000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 47000000000000.0)
   (/ 2.0 (* (* (* (* t (/ t l)) (* (/ t l) (sin k_m))) (tan k_m)) 2.0))
   (*
    (*
     (/ (* l l) (* k_m k_m))
     (/ (cos k_m) (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t)))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5 - (0.5 * cos((2.0 * k_m)))) * t))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 47000000000000.0d0) then
        tmp = 2.0d0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0d0)
    else
        tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 47000000000000.0) {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * Math.sin(k_m))) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (Math.cos(k_m) / ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 47000000000000.0:
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * math.sin(k_m))) * math.tan(k_m)) * 2.0)
	else:
		tmp = (((l * l) / (k_m * k_m)) * (math.cos(k_m) / ((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 47000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k_m))) * tan(k_m)) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 47000000000000.0)
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	else
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5 - (0.5 * cos((2.0 * k_m)))) * t))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 47000000000000.0], N[(2.0 / N[(N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7e13

   1. Initial program 63.3%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 4.7e13 < k

   1. Initial program 46.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 66.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 66.4

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

   6. Applied rewrites 66.4%

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

Alternative 7: 62.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) \cdot k\_m, k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\_m\right)\right) \cdot \tan k\_m\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-29)
   (/
    2.0
    (*
     (/
      (*
       (fma (* (fma (* t t) -0.6666666666666666 1.0) k_m) k_m (* (* t t) 2.0))
       (* k_m k_m))
      (* (cos k_m) l))
     (/ t l)))
   (/ 2.0 (* (* (* (* t (/ t l)) (* (/ t l) (sin k_m))) (tan k_m)) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-29) {
		tmp = 2.0 / (((fma((fma((t * t), -0.6666666666666666, 1.0) * k_m), k_m, ((t * t) * 2.0)) * (k_m * k_m)) / (cos(k_m) * l)) * (t / l));
	} else {
		tmp = 2.0 / ((((t * (t / l)) * ((t / l) * sin(k_m))) * tan(k_m)) * 2.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(Float64(t * t), -0.6666666666666666, 1.0) * k_m), k_m, Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / l)) * Float64(Float64(t / l) * sin(k_m))) * tan(k_m)) * 2.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-29], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.39999999999999992e-29

   1. Initial program 50.5%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 73.5%

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 49.9

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

   9. Applied rewrites 49.9%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 55.9%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 2.39999999999999992e-29 < t

   1. Initial program 67.1%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 69.0

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

   6. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 78.1

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

   8. Applied rewrites 78.1%

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

Alternative 8: 63.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) \cdot k\_m, k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.05e-29)
   (/
    2.0
    (*
     (/
      (*
       (fma (* (fma (* t t) -0.6666666666666666 1.0) k_m) k_m (* (* t t) 2.0))
       (* k_m k_m))
      (* (cos k_m) l))
     (/ t l)))
   (if (<= t 1.15e+63)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-29) {
		tmp = 2.0 / (((fma((fma((t * t), -0.6666666666666666, 1.0) * k_m), k_m, ((t * t) * 2.0)) * (k_m * k_m)) / (cos(k_m) * l)) * (t / l));
	} else if (t <= 1.15e+63) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.05e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(Float64(t * t), -0.6666666666666666, 1.0) * k_m), k_m, Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	elseif (t <= 1.15e+63)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.05e-29], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e+63], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.04999999999999995e-29

   1. Initial program 50.5%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 73.5%

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 49.9

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

   9. Applied rewrites 49.9%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   11. Applied rewrites 55.9%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 1.04999999999999995e-29 < t < 1.14999999999999997e63

   1. Initial program 76.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 65.3

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

   4. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-/r* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lift-pow.f64 74.8

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

   8. Applied rewrites 74.8%

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

   if 1.14999999999999997e63 < t

   1. Initial program 63.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 88.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

Alternative 9: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.3e-93)
   (* (* (/ (* l l) (* k_m k_m)) (/ (cos k_m) (* (* k_m k_m) t))) 2.0)
   (if (<= t 1.15e+63)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	} else if (t <= 1.15e+63) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.3d-93) then
        tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0d0
    else if (t <= 1.15d+63) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / l) * 2.0d0) * (t / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (Math.cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	} else if (t <= 1.15e+63) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.3e-93:
		tmp = (((l * l) / (k_m * k_m)) * (math.cos(k_m) / ((k_m * k_m) * t))) * 2.0
	elif t <= 1.15e+63:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.3e-93)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64(Float64(k_m * k_m) * t))) * 2.0);
	elseif (t <= 1.15e+63)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.3e-93)
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	elseif (t <= 1.15e+63)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / l) * 2.0) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.3e-93], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 1.15e+63], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999998e-93

   1. Initial program 49.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 57.5

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

   7. Applied rewrites 57.5%

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

   if 2.2999999999999998e-93 < t < 1.14999999999999997e63

   1. Initial program 73.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.7

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

   4. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-/r* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lift-pow.f64 71.8

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

   8. Applied rewrites 71.8%

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

   if 1.14999999999999997e63 < t

   1. Initial program 63.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 88.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

Alternative 10: 63.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{{k\_m}^{-2}}{t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.3e-93)
   (* (* (/ (* l l) (* k_m k_m)) (/ (pow k_m -2.0) t)) 2.0)
   (if (<= t 1.15e+63)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (pow(k_m, -2.0) / t)) * 2.0;
	} else if (t <= 1.15e+63) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.3d-93) then
        tmp = (((l * l) / (k_m * k_m)) * ((k_m ** (-2.0d0)) / t)) * 2.0d0
    else if (t <= 1.15d+63) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / l) * 2.0d0) * (t / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (Math.pow(k_m, -2.0) / t)) * 2.0;
	} else if (t <= 1.15e+63) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.3e-93:
		tmp = (((l * l) / (k_m * k_m)) * (math.pow(k_m, -2.0) / t)) * 2.0
	elif t <= 1.15e+63:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.3e-93)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64((k_m ^ -2.0) / t)) * 2.0);
	elseif (t <= 1.15e+63)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.3e-93)
		tmp = (((l * l) / (k_m * k_m)) * ((k_m ^ -2.0) / t)) * 2.0;
	elseif (t <= 1.15e+63)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / l) * 2.0) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.3e-93], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 1.15e+63], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999998e-93

   1. Initial program 49.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 55.0

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

   7. Applied rewrites 55.0%

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

   if 2.2999999999999998e-93 < t < 1.14999999999999997e63

   1. Initial program 73.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.7

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

   4. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-/r* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lift-pow.f64 71.8

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

   8. Applied rewrites 71.8%

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

   if 1.14999999999999997e63 < t

   1. Initial program 63.4%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 74.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 88.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

Alternative 11: 60.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{{k\_m}^{-2}}{t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.3e-93)
   (* (* (/ (* l l) (* k_m k_m)) (/ (pow k_m -2.0) t)) 2.0)
   (if (<= t 5.8e+58)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) (* l l)) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (pow(k_m, -2.0) / t)) * 2.0;
	} else if (t <= 5.8e+58) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.3d-93) then
        tmp = (((l * l) / (k_m * k_m)) * ((k_m ** (-2.0d0)) / t)) * 2.0d0
    else if (t <= 5.8d+58) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / (l * l)) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (Math.pow(k_m, -2.0) / t)) * 2.0;
	} else if (t <= 5.8e+58) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.3e-93:
		tmp = (((l * l) / (k_m * k_m)) * (math.pow(k_m, -2.0) / t)) * 2.0
	elif t <= 5.8e+58:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / (l * l)) * 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.3e-93)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64((k_m ^ -2.0) / t)) * 2.0);
	elseif (t <= 5.8e+58)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.3e-93)
		tmp = (((l * l) / (k_m * k_m)) * ((k_m ^ -2.0) / t)) * 2.0;
	elseif (t <= 5.8e+58)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / (l * l)) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.3e-93], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 5.8e+58], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999998e-93

   1. Initial program 49.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 55.0

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

   7. Applied rewrites 55.0%

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

   if 2.2999999999999998e-93 < t < 5.80000000000000004e58

   1. Initial program 73.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.3

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

   4. Applied rewrites 61.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 63.6

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

   6. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-/r* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lift-pow.f64 71.7

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

   8. Applied rewrites 71.7%

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

   if 5.80000000000000004e58 < t

   1. Initial program 63.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 73.3

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

   7. Applied rewrites 73.3%

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

Alternative 12: 61.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-93}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{{k\_m}^{-2}}{t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.3e-93)
   (* (* (/ (* l l) (* k_m k_m)) (/ (pow k_m -2.0) t)) 2.0)
   (if (<= t 2.55e+64)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ (* l l) (* (pow (* k_m t) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (pow(k_m, -2.0) / t)) * 2.0;
	} else if (t <= 2.55e+64) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.3d-93) then
        tmp = (((l * l) / (k_m * k_m)) * ((k_m ** (-2.0d0)) / t)) * 2.0d0
    else if (t <= 2.55d+64) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = (l * l) / (((k_m * t) ** 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-93) {
		tmp = (((l * l) / (k_m * k_m)) * (Math.pow(k_m, -2.0) / t)) * 2.0;
	} else if (t <= 2.55e+64) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = (l * l) / (Math.pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.3e-93:
		tmp = (((l * l) / (k_m * k_m)) * (math.pow(k_m, -2.0) / t)) * 2.0
	elif t <= 2.55e+64:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = (l * l) / (math.pow((k_m * t), 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.3e-93)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64((k_m ^ -2.0) / t)) * 2.0);
	elseif (t <= 2.55e+64)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.3e-93)
		tmp = (((l * l) / (k_m * k_m)) * ((k_m ^ -2.0) / t)) * 2.0;
	elseif (t <= 2.55e+64)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = (l * l) / (((k_m * t) ^ 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.3e-93], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 2.55e+64], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999998e-93

   1. Initial program 49.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 55.0

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

   7. Applied rewrites 55.0%

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

   if 2.2999999999999998e-93 < t < 2.55000000000000012e64

   1. Initial program 74.0%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.7

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

   4. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 63.9

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

   6. Applied rewrites 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-/r* N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. lower-/.f64 N/A

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

      23. lift-pow.f64 71.8

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

   8. Applied rewrites 71.8%

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

   if 2.55000000000000012e64 < t

   1. Initial program 63.2%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 52.9

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

   4. Applied rewrites 52.9%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 52.9

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

   6. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 73.6

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   8. Applied rewrites 73.6%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k\_m}^{4} \cdot t} \cdot 2\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.2e-93)
   (* (/ (* l l) (* (pow k_m 4.0) t)) 2.0)
   (if (<= t 2.55e+64)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ (* l l) (* (pow (* k_m t) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.2e-93) {
		tmp = ((l * l) / (pow(k_m, 4.0) * t)) * 2.0;
	} else if (t <= 2.55e+64) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.2d-93) then
        tmp = ((l * l) / ((k_m ** 4.0d0) * t)) * 2.0d0
    else if (t <= 2.55d+64) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = (l * l) / (((k_m * t) ** 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.2e-93) {
		tmp = ((l * l) / (Math.pow(k_m, 4.0) * t)) * 2.0;
	} else if (t <= 2.55e+64) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = (l * l) / (Math.pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.2e-93:
		tmp = ((l * l) / (math.pow(k_m, 4.0) * t)) * 2.0
	elif t <= 2.55e+64:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = (l * l) / (math.pow((k_m * t), 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.2e-93)
		tmp = Float64(Float64(Float64(l * l) / Float64((k_m ^ 4.0) * t)) * 2.0);
	elseif (t <= 2.55e+64)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.2e-93)
		tmp = ((l * l) / ((k_m ^ 4.0) * t)) * 2.0;
	elseif (t <= 2.55e+64)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = (l * l) / (((k_m * t) ^ 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.2e-93], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 2.55e+64], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.19999999999999996e-93

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      5. lower-pow.f64 53.2

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. Applied rewrites 53.2%

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   if 2.19999999999999996e-93 < t < 2.55000000000000012e64

   1. Initial program 74.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 61.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 63.9

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. Applied rewrites 63.9%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. pow2 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{\color{blue}{t}}^{3}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      17. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{\color{blue}{t}}^{3}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      19. times-frac N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

      21. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

      23. lift-pow.f64 71.8

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   8. Applied rewrites 71.8%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 2.55000000000000012e64 < t

   1. Initial program 63.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 52.9

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 52.9

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 52.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 73.6

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   8. Applied rewrites 73.6%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}{\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot \ell, -0.5, \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k\_m}^{4} \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.1e-25)
   (/ (* l l) (* (pow (* k_m t) 2.0) t))
   (if (<= k_m 1.35e+61)
     (/
      2.0
      (/
       (*
        (*
         (fma
          (fma -0.6666666666666666 (* t t) 1.0)
          (* k_m k_m)
          (* (* t t) 2.0))
         (* k_m k_m))
        t)
       (* (fma (* (* k_m k_m) l) -0.5 l) l)))
     (* (/ (* l l) (* (pow k_m 4.0) t)) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-25) {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	} else if (k_m <= 1.35e+61) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) * t) / (fma(((k_m * k_m) * l), -0.5, l) * l));
	} else {
		tmp = ((l * l) / (pow(k_m, 4.0) * t)) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.1e-25)
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	elseif (k_m <= 1.35e+61)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) * t) / Float64(fma(Float64(Float64(k_m * k_m) * l), -0.5, l) * l)));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64((k_m ^ 4.0) * t)) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.1e-25], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.35e+61], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] * -0.5 + l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.1000000000000003e-25

   1. Initial program 64.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 57.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 57.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 75.2

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   8. Applied rewrites 75.2%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]

   if 5.1000000000000003e-25 < k < 1.3500000000000001e61

   1. Initial program 55.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   4. Applied rewrites 83.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Applied rewrites 82.9%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right) + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) \cdot {k}^{2} + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      17. lift-*.f64 43.5

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 43.5%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\ell + \frac{-1}{2} \cdot \left({k}^{2} \cdot \ell\right)\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\frac{-1}{2} \cdot \left({k}^{2} \cdot \ell\right) + \ell\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\left({k}^{2} \cdot \ell\right) \cdot \frac{-1}{2} + \ell\right) \cdot \ell}} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left({k}^{2} \cdot \ell, \frac{-1}{2}, \ell\right) \cdot \ell}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left({k}^{2} \cdot \ell, \frac{-1}{2}, \ell\right) \cdot \ell}} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, \frac{-1}{2}, \ell\right) \cdot \ell}} \]

       6. lift-*.f64 39.4

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, -0.5, \ell\right) \cdot \ell}} \]

   12. Applied rewrites 39.4%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, -0.5, \ell\right) \cdot \ell}} \]

   if 1.3500000000000001e61 < k

   1. Initial program 45.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      5. lower-pow.f64 55.5

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-20}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot {t}^{3}\right)}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}{\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot \ell, -0.5, \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k\_m}^{4} \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-20)
   (* l (/ l (* k_m (* k_m (pow t 3.0)))))
   (if (<= k_m 1.35e+61)
     (/
      2.0
      (/
       (*
        (*
         (fma
          (fma -0.6666666666666666 (* t t) 1.0)
          (* k_m k_m)
          (* (* t t) 2.0))
         (* k_m k_m))
        t)
       (* (fma (* (* k_m k_m) l) -0.5 l) l)))
     (* (/ (* l l) (* (pow k_m 4.0) t)) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-20) {
		tmp = l * (l / (k_m * (k_m * pow(t, 3.0))));
	} else if (k_m <= 1.35e+61) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) * t) / (fma(((k_m * k_m) * l), -0.5, l) * l));
	} else {
		tmp = ((l * l) / (pow(k_m, 4.0) * t)) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e-20)
		tmp = Float64(l * Float64(l / Float64(k_m * Float64(k_m * (t ^ 3.0)))));
	elseif (k_m <= 1.35e+61)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) * t) / Float64(fma(Float64(Float64(k_m * k_m) * l), -0.5, l) * l)));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64((k_m ^ 4.0) * t)) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e-20], N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.35e+61], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] * -0.5 + l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.00000000000000029e-20

   1. Initial program 63.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 57.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 63.8

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. Applied rewrites 63.8%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{{t}^{3}}\right)} \]

      7. lift-pow.f64 73.5

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{\color{blue}{3}}\right)} \]

   8. Applied rewrites 73.5%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   if 3.00000000000000029e-20 < k < 1.3500000000000001e61

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   4. Applied rewrites 83.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Applied rewrites 83.1%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right) + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) \cdot {k}^{2} + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      17. lift-*.f64 42.5

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 42.5%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\ell + \frac{-1}{2} \cdot \left({k}^{2} \cdot \ell\right)\right) \cdot \ell}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\frac{-1}{2} \cdot \left({k}^{2} \cdot \ell\right) + \ell\right) \cdot \ell}} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\left({k}^{2} \cdot \ell\right) \cdot \frac{-1}{2} + \ell\right) \cdot \ell}} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left({k}^{2} \cdot \ell, \frac{-1}{2}, \ell\right) \cdot \ell}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left({k}^{2} \cdot \ell, \frac{-1}{2}, \ell\right) \cdot \ell}} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, \frac{-1}{2}, \ell\right) \cdot \ell}} \]

       6. lift-*.f64 38.2

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, -0.5, \ell\right) \cdot \ell}} \]

   12. Applied rewrites 38.2%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \ell, -0.5, \ell\right) \cdot \ell}} \]

   if 1.3500000000000001e61 < k

   1. Initial program 45.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

      5. lower-pow.f64 55.5

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   7. Applied rewrites 55.5%

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot {t}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.25e+142)
   (/
    2.0
    (/
     (*
      (*
       (fma (fma -0.6666666666666666 (* t t) 1.0) (* k_m k_m) (* (* t t) 2.0))
       (* k_m k_m))
      t)
     (* l l)))
   (* l (/ l (* k_m (* k_m (pow t 3.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.25e+142) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) * t) / (l * l));
	} else {
		tmp = l * (l / (k_m * (k_m * pow(t, 3.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.25e+142)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) * t) / Float64(l * l)));
	else
		tmp = Float64(l * Float64(l / Float64(k_m * Float64(k_m * (t ^ 3.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.25e+142], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(k$95$m * N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e142

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   4. Applied rewrites 73.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Applied rewrites 73.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right) + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) \cdot {k}^{2} + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      17. lift-*.f64 51.8

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 51.8%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}} \]
   11. Step-by-step derivation

   12. Applied rewrites 50.7%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}} \]

   if 1.25e142 < t

   1. Initial program 66.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 54.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 60.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. Applied rewrites 60.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      4. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{{t}^{3}}\right)} \]

      7. lift-pow.f64 72.0

         \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot {t}^{\color{blue}{3}}\right)} \]

   8. Applied rewrites 72.0%

      \[\leadsto \ell \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.25e+142)
   (/
    2.0
    (/
     (*
      (*
       (fma (fma -0.6666666666666666 (* t t) 1.0) (* k_m k_m) (* (* t t) 2.0))
       (* k_m k_m))
      t)
     (* l l)))
   (* l (/ l (* (* k_m k_m) (* (* t t) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.25e+142) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) * t) / (l * l));
	} else {
		tmp = l * (l / ((k_m * k_m) * ((t * t) * t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.25e+142)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) * t) / Float64(l * l)));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.25e+142], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e142

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   4. Applied rewrites 73.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Applied rewrites 73.4%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right)\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left({k}^{2} \cdot \left(1 + \frac{-2}{3} \cdot {t}^{2}\right) + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) \cdot {k}^{2} + 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(1 + \frac{-2}{3} \cdot {t}^{2}, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\frac{-2}{3} \cdot {t}^{2} + 1, {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, {t}^{2}, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), {k}^{2}, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, 2 \cdot {t}^{2}\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, {t}^{2} \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot {k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      16. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      17. lift-*.f64 51.8

          \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 51.8%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{3}, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}} \]
   11. Step-by-step derivation

   12. Applied rewrites 50.7%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell \cdot \ell}} \]

   if 1.25e142 < t

   1. Initial program 66.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 54.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 54.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 60.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. Applied rewrites 60.3%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lift-*.f64 60.3

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   8. Applied rewrites 60.3%

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 55.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* (* k_m k_m) (* (* t t) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / ((k_m * k_m) * ((t * t) * t)));
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / ((k_m * k_m) * ((t * t) * t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / ((k_m * k_m) * ((t * t) * t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / ((k_m * k_m) * ((t * t) * t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / ((k_m * k_m) * ((t * t) * t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 51.2

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   3. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   7. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

   13. lift-*.f64 55.5

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

6. Applied rewrites 55.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. unpow3 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

   5. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lift-*.f64 55.5

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

8. Applied rewrites 55.5%

   \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))