Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 77.0%

Time: 6.8s

Alternatives: 17

Speedup: 6.5×

• 26.8% 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: 54.5% 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: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{\cos k \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \frac{1}{\frac{\cos k}{\sin k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.8e-151)
    (*
     (/
      (* (cos k) (* l_m l_m))
      (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ k k)))))))
     2.0)
    (/
     2.0
     (*
      (*
       (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k))
       (/ 1.0 (/ (cos k) (sin k))))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.8e-151) {
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * (1.0 / (cos(k) / sin(k)))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9.8d-151) then
        tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5d0 - (0.5d0 * cos((k + k))))))) * 2.0d0
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * (1.0d0 / (cos(k) / sin(k)))) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.8e-151) {
		tmp = ((Math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * Math.cos((k + k))))))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * (1.0 / (Math.cos(k) / Math.sin(k)))) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9.8e-151:
		tmp = ((math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * math.cos((k + k))))))) * 2.0
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * (1.0 / (math.cos(k) / math.sin(k)))) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9.8e-151)
		tmp = Float64(Float64(Float64(cos(k) * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * Float64(1.0 / Float64(cos(k) / sin(k)))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9.8e-151)
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * (1.0 / (cos(k) / sin(k)))) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.8e-151], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.79999999999999933e-151

   1. Initial program 27.5%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 27.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 69.6

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 69.6

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

   7. Applied rewrites 69.6%

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

   if 9.79999999999999933e-151 < t

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 79.5

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

   3. Applied rewrites 79.5%

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

      1. lift-tan.f64 N/A

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

      2. quot-tan N/A

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

      3. division-flip N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-sin.f64 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 2: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{\cos k \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{1}{\frac{t\_m}{k}}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.8e-151)
    (*
     (/
      (* (cos k) (* l_m l_m))
      (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ k k)))))))
     2.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      (+ (+ 1.0 (pow (/ 1.0 (/ t_m k)) 2.0)) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.8e-151) {
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + pow((1.0 / (t_m / k)), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9.8d-151) then
        tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5d0 - (0.5d0 * cos((k + k))))))) * 2.0d0
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * ((1.0d0 + ((1.0d0 / (t_m / k)) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.8e-151) {
		tmp = ((Math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * Math.cos((k + k))))))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((1.0 / (t_m / k)), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9.8e-151:
		tmp = ((math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * math.cos((k + k))))))) * 2.0
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((1.0 / (t_m / k)), 2.0)) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9.8e-151)
		tmp = Float64(Float64(Float64(cos(k) * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(1.0 / Float64(t_m / k)) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9.8e-151)
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + ((1.0 / (t_m / k)) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.8e-151], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(1.0 / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.79999999999999933e-151

   1. Initial program 27.5%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 27.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 69.6

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 69.6

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

   7. Applied rewrites 69.6%

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

   if 9.79999999999999933e-151 < t

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 79.5

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

   3. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. division-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 3: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{\cos k \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.8e-151)
    (*
     (/
      (* (cos k) (* l_m l_m))
      (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ k k)))))))
     2.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      (+ (+ 1.0 (* (/ k t_m) (/ k t_m))) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.8e-151) {
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9.8d-151) then
        tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5d0 - (0.5d0 * cos((k + k))))))) * 2.0d0
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) * (k / t_m))) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 9.8e-151) {
		tmp = ((Math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * Math.cos((k + k))))))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 9.8e-151:
		tmp = ((math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * math.cos((k + k))))))) * 2.0
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 9.8e-151)
		tmp = Float64(Float64(Float64(cos(k) * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k / t_m) * Float64(k / t_m))) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 9.8e-151)
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) * (k / t_m))) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.8e-151], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.79999999999999933e-151

   1. Initial program 27.5%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 27.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 69.6

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 69.6

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

   7. Applied rewrites 69.6%

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

   if 9.79999999999999933e-151 < t

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 79.5

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

   3. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 4: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.6e+20)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      2.0))
    (*
     (/
      (* (cos k) (* l_m l_m))
      (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ k k)))))))
     2.0))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 4.6e+20) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.6d+20) then
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5d0 - (0.5d0 * cos((k + k))))))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 4.6e+20) {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = ((Math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * Math.cos((k + k))))))) * 2.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 4.6e+20:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = ((math.cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * math.cos((k + k))))))) * 2.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 4.6e+20)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(cos(k) * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 4.6e+20)
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 4.6e+20], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.6e20

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in t around inf

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

   6. Applied rewrites 68.3%

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

   if 4.6e20 < k

   1. Initial program 47.3%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 68.0

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

Alternative 5: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-132}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 52000000000:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-132)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) k) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= k 52000000000.0)
      (/ 1.0 (* (* k k) (exp (fma 3.0 (log t_m) (* -2.0 (log l_m))))))
      (*
       (/
        (* (cos k) (* l_m l_m))
        (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ k k)))))))
       2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1e-132) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * k) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 52000000000.0) {
		tmp = 1.0 / ((k * k) * exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))));
	} else {
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1e-132)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * k) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (k <= 52000000000.0)
		tmp = Float64(1.0 / Float64(Float64(k * k) * exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(cos(k) * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1e-132], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 52000000000.0], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 9.9999999999999999e-133

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 72.4

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 67.5%

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

   if 9.9999999999999999e-133 < k < 5.2e10

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 73.4

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   5. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      2. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 73.1%

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

   if 5.2e10 < k

   1. Initial program 47.8%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 68.0

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

Alternative 6: 63.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\\ \mathbf{elif}\;k \leq 52000000000:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(l\_m \cdot l\_m\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.3e-130)
    (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))
    (if (<= k 52000000000.0)
      (/ 1.0 (* (* k k) (exp (fma 3.0 (log t_m) (* -2.0 (log l_m))))))
      (*
       (/
        (* (cos k) (* l_m l_m))
        (* (* k k) (* t_m (- 0.5 (* 0.5 (cos (+ k k)))))))
       2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.3e-130) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else if (k <= 52000000000.0) {
		tmp = 1.0 / ((k * k) * exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))));
	} else {
		tmp = ((cos(k) * (l_m * l_m)) / ((k * k) * (t_m * (0.5 - (0.5 * cos((k + k))))))) * 2.0;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.3e-130)
		tmp = Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m)));
	elseif (k <= 52000000000.0)
		tmp = Float64(1.0 / Float64(Float64(k * k) * exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(cos(k) * Float64(l_m * l_m)) / Float64(Float64(k * k) * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.3e-130], N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 52000000000.0], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.2999999999999998e-130

   1. Initial program 56.4%

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

   2. Taylor expanded in 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. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow3 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if 3.2999999999999998e-130 < k < 5.2e10

   1. Initial program 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 73.6

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

   3. Applied rewrites 73.6%

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

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   5. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      2. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 73.3%

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

   if 5.2e10 < k

   1. Initial program 47.8%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 68.0

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 68.0

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

   7. Applied rewrites 68.0%

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

Alternative 7: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{k \cdot \left(k \cdot \left(t\_m \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.3e-130)
    (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))
    (if (<= k 3e+35)
      (/ 1.0 (* (* k k) (exp (fma 3.0 (log t_m) (* -2.0 (log l_m))))))
      (*
       (/ (* l_m l_m) (* k (* k (* t_m (- 0.5 (* 0.5 (cos (+ k k))))))))
       2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.3e-130) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else if (k <= 3e+35) {
		tmp = 1.0 / ((k * k) * exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))));
	} else {
		tmp = ((l_m * l_m) / (k * (k * (t_m * (0.5 - (0.5 * cos((k + k)))))))) * 2.0;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.3e-130)
		tmp = Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m)));
	elseif (k <= 3e+35)
		tmp = Float64(1.0 / Float64(Float64(k * k) * exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(l_m * l_m) / Float64(k * Float64(k * Float64(t_m * Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))))))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.3e-130], N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3e+35], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * N[(k * N[(t$95$m * N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.2999999999999998e-130

   1. Initial program 56.4%

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

   2. Taylor expanded in 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. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow3 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if 3.2999999999999998e-130 < k < 2.99999999999999991e35

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 73.0

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   5. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      2. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 69.1%

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

   if 2.99999999999999991e35 < k

   1. Initial program 47.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 43.2

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

   7. Applied rewrites 43.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 55.8

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 55.8

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

   10. Applied rewrites 55.8%

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

   12. Applied rewrites 57.0%

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

Alternative 8: 61.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.3e-130)
    (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))
    (if (<= k 3.1e+35)
      (/ 1.0 (* (* k k) (exp (fma 3.0 (log t_m) (* -2.0 (log l_m))))))
      (* (/ (* l_m l_m) (* (* k k) (* t_m (* k k)))) 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.3e-130) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else if (k <= 3.1e+35) {
		tmp = 1.0 / ((k * k) * exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))));
	} else {
		tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.3e-130)
		tmp = Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m)));
	elseif (k <= 3.1e+35)
		tmp = Float64(1.0 / Float64(Float64(k * k) * exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m))))));
	else
		tmp = Float64(Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(t_m * Float64(k * k)))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.3e-130], N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1e+35], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.2999999999999998e-130

   1. Initial program 56.4%

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

   2. Taylor expanded in 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. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow3 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if 3.2999999999999998e-130 < k < 3.09999999999999987e35

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 73.0

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   5. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      2. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 69.1%

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

   if 3.09999999999999987e35 < k

   1. Initial program 47.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 43.2

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

   7. Applied rewrites 43.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 55.8

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 55.8

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

   10. Applied rewrites 55.8%

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

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

Alternative 9: 60.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(-2, \log l\_m, 3 \cdot \log t\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.3e-130)
    (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))
    (if (<= k 3.1e+35)
      (/ 1.0 (* (* k k) (exp (fma -2.0 (log l_m) (* 3.0 (log t_m))))))
      (* (/ (* l_m l_m) (* (* k k) (* t_m (* k k)))) 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.3e-130) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else if (k <= 3.1e+35) {
		tmp = 1.0 / ((k * k) * exp(fma(-2.0, log(l_m), (3.0 * log(t_m)))));
	} else {
		tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.3e-130)
		tmp = Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m)));
	elseif (k <= 3.1e+35)
		tmp = Float64(1.0 / Float64(Float64(k * k) * exp(fma(-2.0, log(l_m), Float64(3.0 * log(t_m))))));
	else
		tmp = Float64(Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(t_m * Float64(k * k)))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.3e-130], N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1e+35], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[Exp[N[(-2.0 * N[Log[l$95$m], $MachinePrecision] + N[(3.0 * N[Log[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.2999999999999998e-130

   1. Initial program 56.4%

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

   2. Taylor expanded in 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. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow3 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.7

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

   8. Applied rewrites 60.7%

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

   if 3.2999999999999998e-130 < k < 3.09999999999999987e35

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 73.0

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   5. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      2. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 69.1%

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

   7. Taylor expanded in t around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-log.f64 69.1

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

   9. Applied rewrites 69.1%

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

   if 3.09999999999999987e35 < k

   1. Initial program 47.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 43.2

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

   7. Applied rewrites 43.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 55.8

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 55.8

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

   10. Applied rewrites 55.8%

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

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

Alternative 10: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot t\_2} \leq \infty:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}{l\_m \cdot l\_m} \cdot k\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(3, \log t\_m, -2 \cdot \log l\_m\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k)) t_2))
         INFINITY)
      (/ 2.0 (* (* (/ (* k (* (* t_m t_m) t_m)) (* l_m l_m)) k) t_2))
      (/ 1.0 (* (* k k) (exp (fma 3.0 (log t_m) (* -2.0 (log l_m))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= ((double) INFINITY)) {
		tmp = 2.0 / ((((k * ((t_m * t_m) * t_m)) / (l_m * l_m)) * k) * t_2);
	} else {
		tmp = 1.0 / ((k * k) * exp(fma(3.0, log(t_m), (-2.0 * log(l_m)))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * t_2)) <= Inf)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * Float64(Float64(t_m * t_m) * t_m)) / Float64(l_m * l_m)) * k) * t_2));
	else
		tmp = Float64(1.0 / Float64(Float64(k * k) * exp(fma(3.0, log(t_m), Float64(-2.0 * log(l_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := Block[{t$95$2 = N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 / N[(N[(N[(N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[Exp[N[(3.0 * N[Log[t$95$m], $MachinePrecision] + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 82.0%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 78.2

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

   7. Applied rewrites 78.2%

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

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 38.8

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

   3. Applied rewrites 38.8%

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

   4. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}}} \]
   5. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      2. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      4. pow2 N/A

         \[\leadsto \frac{1}{{k}^{2} \cdot e^{3 \cdot \log t - 2 \cdot \log \ell}} \]

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\frac{1}{\left(k \cdot k\right) \cdot e^{\mathsf{fma}\left(3, \log t, -2 \cdot \log \ell\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.1e+35)
    (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))
    (* (/ (* l_m l_m) (* (* k k) (* t_m (* k k)))) 2.0))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.1e+35) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else {
		tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d+35) then
        tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m))
    else
        tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 3.1e+35) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else {
		tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 3.1e+35:
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m))
	else:
		tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 3.1e+35)
		tmp = Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(t_m * Float64(k * k)))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 3.1e+35)
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	else
		tmp = ((l_m * l_m) / ((k * k) * (t_m * (k * k)))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 3.1e+35], N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999987e35

   1. Initial program 56.7%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 52.7

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

   4. Applied rewrites 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow3 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 57.2

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 61.2

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

   8. Applied rewrites 61.2%

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

   if 3.09999999999999987e35 < k

   1. Initial program 47.0%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 43.2

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

   7. Applied rewrites 43.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 55.8

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 55.8

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

   10. Applied rewrites 55.8%

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

   11. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

Alternative 12: 59.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+188}:\\ \;\;\;\;\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e+188)
    (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))
    (* (/ (* l_m l_m) (* (* (* k k) (* k k)) t_m)) 2.0))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2e+188) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else {
		tmp = ((l_m * l_m) / (((k * k) * (k * k)) * t_m)) * 2.0;
	}
	return t_s * tmp;
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2d+188) then
        tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m))
    else
        tmp = ((l_m * l_m) / (((k * k) * (k * k)) * t_m)) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2e+188) {
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	} else {
		tmp = ((l_m * l_m) / (((k * k) * (k * k)) * t_m)) * 2.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2e+188:
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m))
	else:
		tmp = ((l_m * l_m) / (((k * k) * (k * k)) * t_m)) * 2.0
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2e+188)
		tmp = Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(Float64(Float64(l_m * l_m) / Float64(Float64(Float64(k * k) * Float64(k * k)) * t_m)) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2e+188)
		tmp = ((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m));
	else
		tmp = ((l_m * l_m) / (((k * k) * (k * k)) * t_m)) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2e+188], N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2e188

   1. Initial program 55.4%

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

   2. Taylor expanded in 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. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow3 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 55.5

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

   6. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if 2e188 < k

   1. Initial program 46.6%

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

   2. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 46.7

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 60.1

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 60.1

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

   10. Applied rewrites 60.1%

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

   11. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. sqr-pow N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 60.3

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

   13. Applied rewrites 60.3%

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

Alternative 13: 58.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\frac{l\_m}{k}}{k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* (/ (/ l_m k) k) (/ l_m (* (* t_m t_m) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m)));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(Float64(l_m / k) / k) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (((l_m / k) / k) * (l_m / ((t_m * t_m) * t_m)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(N[(N[(l$95$m / k), $MachinePrecision] / k), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

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. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. pow3 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow3 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 54.9

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

6. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 58.3

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

8. Applied rewrites 58.3%

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

Alternative 14: 57.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{l\_m}{k \cdot k} \cdot \frac{\frac{l\_m}{t\_m \cdot t\_m}}{t\_m}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* (/ l_m (* k k)) (/ (/ l_m (* t_m t_m)) t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m / (k * k)) * ((l_m / (t_m * t_m)) / t_m));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * ((l_m / (k * k)) * ((l_m / (t_m * t_m)) / t_m))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m / (k * k)) * ((l_m / (t_m * t_m)) / t_m));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * ((l_m / (k * k)) * ((l_m / (t_m * t_m)) / t_m))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(l_m / Float64(k * k)) * Float64(Float64(l_m / Float64(t_m * t_m)) / t_m)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * ((l_m / (k * k)) * ((l_m / (t_m * t_m)) / t_m));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(N[(l$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

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. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. pow3 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow3 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 54.9

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

6. Applied rewrites 54.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. unpow3 N/A

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

   6. pow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 57.2

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

8. Applied rewrites 57.2%

   \[\leadsto \frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t \cdot t}}{\color{blue}{t}} \]
9. Add Preprocessing

Alternative 15: 55.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{l\_m}{k \cdot k} \cdot l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ (* (/ l_m (* k k)) l_m) (* (* t_m t_m) t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (((l_m / (k * k)) * l_m) / ((t_m * t_m) * t_m));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (((l_m / (k * k)) * l_m) / ((t_m * t_m) * t_m))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (((l_m / (k * k)) * l_m) / ((t_m * t_m) * t_m));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (((l_m / (k * k)) * l_m) / ((t_m * t_m) * t_m))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(Float64(l_m / Float64(k * k)) * l_m) / Float64(Float64(t_m * t_m) * t_m)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (((l_m / (k * k)) * l_m) / ((t_m * t_m) * t_m));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(N[(N[(l$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

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. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. pow3 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow3 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 54.9

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

6. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow3 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. pow3 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 54.4

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

8. Applied rewrites 54.4%

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

Alternative 16: 54.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{l\_m}{k \cdot k} \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* (/ l_m (* k k)) (/ l_m (* (* t_m t_m) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m / (k * k)) * (l_m / ((t_m * t_m) * t_m)));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * ((l_m / (k * k)) * (l_m / ((t_m * t_m) * t_m)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m / (k * k)) * (l_m / ((t_m * t_m) * t_m)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * ((l_m / (k * k)) * (l_m / ((t_m * t_m) * t_m)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(l_m / Float64(k * k)) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * ((l_m / (k * k)) * (l_m / ((t_m * t_m) * t_m)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(N[(l$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

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. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 50.8

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

4. Applied rewrites 50.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. pow3 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. pow3 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 54.9

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

6. Applied rewrites 54.9%

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

Alternative 17: 54.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* l_m (/ l_m (* (* k k) (* (* t_m t_m) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
}

Fortran

l_m =     private
t\_m =     private
t\_s =     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_s, t_m, l_m, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(l_m * Float64(l_m / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * (l_m * (l_m / ((k * k) * ((t_m * t_m) * t_m))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k_] := N[(t$95$s * N[(l$95$m * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

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. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   11. lower-*.f64 50.8

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   3. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   9. pow2 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   10. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   12. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   13. pow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   17. lift-*.f64 55.1

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

6. Applied rewrites 55.1%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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))))