Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 94.6%

Time: 9.3s

Alternatives: 19

Speedup: 9.6×

• 38.8% 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: 35.7% 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: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \frac{t\_2}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t\_2 \cdot t\_m}\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 4e+44)
      (/ 2.0 (* (* (/ k l) (/ (* k t_m) l)) (/ t_2 (cos k))))
      (* (* (* (/ l k) (/ l k)) (/ (cos k) (* t_2 t_m))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 4e+44) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (t_2 / cos(k)));
	} else {
		tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (t_m <= 4d+44) then
        tmp = 2.0d0 / (((k / l) * ((k * t_m) / l)) * (t_2 / cos(k)))
    else
        tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 4e+44) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (t_2 / Math.cos(k)));
	} else {
		tmp = (((l / k) * (l / k)) * (Math.cos(k) / (t_2 * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 4e+44:
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (t_2 / math.cos(k)))
	else:
		tmp = (((l / k) * (l / k)) * (math.cos(k) / (t_2 * t_m))) * 2.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 4e+44)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(t_2 / cos(k))));
	else
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t_2 * t_m))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 4e+44)
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (t_2 / cos(k)));
	else
		tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e+44], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000004e44

   1. Initial program 37.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 95.7%

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

   if 4.0000000000000004e44 < t

   1. Initial program 23.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 72.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 96.2

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

   12. Applied rewrites 96.2%

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

Alternative 2: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot t\_m\right)\right)}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t\_2 \cdot t\_m}\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= k 6.3e+99)
      (/ 2.0 (/ (* t_2 (* (/ k l) (* k t_m))) (* (cos k) l)))
      (* (* (* (/ l k) (/ l k)) (/ (cos k) (* t_2 t_m))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 6.3e+99) {
		tmp = 2.0 / ((t_2 * ((k / l) * (k * t_m))) / (cos(k) * l));
	} else {
		tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (k <= 6.3d+99) then
        tmp = 2.0d0 / ((t_2 * ((k / l) * (k * t_m))) / (cos(k) * l))
    else
        tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 6.3e+99) {
		tmp = 2.0 / ((t_2 * ((k / l) * (k * t_m))) / (Math.cos(k) * l));
	} else {
		tmp = (((l / k) * (l / k)) * (Math.cos(k) / (t_2 * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 6.3e+99:
		tmp = 2.0 / ((t_2 * ((k / l) * (k * t_m))) / (math.cos(k) * l))
	else:
		tmp = (((l / k) * (l / k)) * (math.cos(k) / (t_2 * t_m))) * 2.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 6.3e+99)
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l) * Float64(k * t_m))) / Float64(cos(k) * l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t_2 * t_m))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 6.3e+99)
		tmp = 2.0 / ((t_2 * ((k / l) * (k * t_m))) / (cos(k) * l));
	else
		tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6.3e+99], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.2999999999999996e99

   1. Initial program 35.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. frac-times N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 93.6%

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

   if 6.2999999999999996e99 < k

   1. Initial program 31.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 59.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 97.2

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

   12. Applied rewrites 97.2%

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

Alternative 3: 91.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot \left(\frac{k}{\ell} \cdot k\right)}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t\_2}\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow (sin k) 2.0) t_m)))
   (*
    t_s
    (if (<= k 6.3e+100)
      (/ 2.0 (/ (* t_2 (* (/ k l) k)) (* (cos k) l)))
      (* (* (* (/ l k) (/ l k)) (/ (cos k) t_2)) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0) * t_m;
	double tmp;
	if (k <= 6.3e+100) {
		tmp = 2.0 / ((t_2 * ((k / l) * k)) / (cos(k) * l));
	} else {
		tmp = (((l / k) * (l / k)) * (cos(k) / t_2)) * 2.0;
	}
	return t_s * tmp;
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (sin(k) ** 2.0d0) * t_m
    if (k <= 6.3d+100) then
        tmp = 2.0d0 / ((t_2 * ((k / l) * k)) / (cos(k) * l))
    else
        tmp = (((l / k) * (l / k)) * (cos(k) / t_2)) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0) * t_m;
	double tmp;
	if (k <= 6.3e+100) {
		tmp = 2.0 / ((t_2 * ((k / l) * k)) / (Math.cos(k) * l));
	} else {
		tmp = (((l / k) * (l / k)) * (Math.cos(k) / t_2)) * 2.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0) * t_m
	tmp = 0
	if k <= 6.3e+100:
		tmp = 2.0 / ((t_2 * ((k / l) * k)) / (math.cos(k) * l))
	else:
		tmp = (((l / k) * (l / k)) * (math.cos(k) / t_2)) * 2.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((sin(k) ^ 2.0) * t_m)
	tmp = 0.0
	if (k <= 6.3e+100)
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(Float64(k / l) * k)) / Float64(cos(k) * l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / t_2)) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (sin(k) ^ 2.0) * t_m;
	tmp = 0.0;
	if (k <= 6.3e+100)
		tmp = 2.0 / ((t_2 * ((k / l) * k)) / (cos(k) * l));
	else
		tmp = (((l / k) * (l / k)) * (cos(k) / t_2)) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6.3e+100], N[(2.0 / N[(N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.3000000000000004e100

   1. Initial program 35.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-cos.f64 92.6

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

   9. Applied rewrites 92.6%

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

   if 6.3000000000000004e100 < k

   1. Initial program 31.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 59.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 97.2

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

   12. Applied rewrites 97.2%

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

Alternative 4: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.059:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t\_m}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.059)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma
        (fma
         (fma 0.03432539682539683 (* k k) 0.08611111111111111)
         (* k k)
         0.16666666666666666)
        (* k k)
        1.0)
       (* k k))))
    (* (* (* (/ l k) (/ l k)) (/ (cos k) (* (pow (sin k) 2.0) t_m))) 2.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.059) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = (((l / k) * (l / k)) * (cos(k) / (pow(sin(k), 2.0) * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.059)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t_m))) * 2.0);
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 0.059], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.058999999999999997

   1. Initial program 37.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.058999999999999997 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 89.2%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 66.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 92.8

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

   12. Applied rewrites 92.8%

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

Alternative 5: 81.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.061:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\_m}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.061)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma
        (fma
         (fma 0.03432539682539683 (* k k) 0.08611111111111111)
         (* k k)
         0.16666666666666666)
        (* k k)
        1.0)
       (* k k))))
    (/
     2.0
     (*
      (/ (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t_m) (cos k))
      (* (/ k l) (/ k l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.061) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((2.0 * k)))) * t_m) / cos(k)) * ((k / l) * (k / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.061)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t_m) / cos(k)) * Float64(Float64(k / l) * Float64(k / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 0.061], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.060999999999999999

   1. Initial program 37.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.060999999999999999 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 93.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 92.9

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

   9. Applied rewrites 92.9%

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

Alternative 6: 81.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.031:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ k l) (/ (* k t_m) l))))
   (*
    t_s
    (if (<= k 0.031)
      (/
       2.0
       (*
        t_2
        (*
         (fma
          (fma
           (fma 0.03432539682539683 (* k k) 0.08611111111111111)
           (* k k)
           0.16666666666666666)
          (* k k)
          1.0)
         (* k k))))
      (/ 2.0 (* t_2 (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) (cos k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k / l) * ((k * t_m) / l);
	double tmp;
	if (k <= 0.031) {
		tmp = 2.0 / (t_2 * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (t_2 * ((0.5 - (0.5 * cos((2.0 * k)))) / cos(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k / l) * Float64(Float64(k * t_m) / l))
	tmp = 0.0
	if (k <= 0.031)
		tmp = Float64(2.0 / Float64(t_2 * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / cos(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := Block[{t$95$2 = N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 0.031], N[(2.0 / N[(t$95$2 * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.031

   1. Initial program 37.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.031 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 89.2

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

   9. Applied rewrites 89.2%

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

Alternative 7: 77.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.059:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.24 \cdot 10^{+147}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\_m}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t\_m} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.059)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma
        (fma
         (fma 0.03432539682539683 (* k k) 0.08611111111111111)
         (* k k)
         0.16666666666666666)
        (* k k)
        1.0)
       (* k k))))
    (if (<= k 1.24e+147)
      (*
       (*
        (/ (* l l) (* k k))
        (/ (cos k) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t_m)))
       2.0)
      (* (/ (pow (/ l k) 2.0) t_m) -0.3333333333333333)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.059) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else if (k <= 1.24e+147) {
		tmp = (((l * l) / (k * k)) * (cos(k) / ((0.5 - (0.5 * cos((2.0 * k)))) * t_m))) * 2.0;
	} else {
		tmp = (pow((l / k), 2.0) / t_m) * -0.3333333333333333;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.059)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	elseif (k <= 1.24e+147)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(cos(k) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t_m))) * 2.0);
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t_m) * -0.3333333333333333);
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 0.059], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.24e+147], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 0.058999999999999997

   1. Initial program 37.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.058999999999999997 < k < 1.2400000000000001e147

   1. Initial program 20.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 80.8%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 80.9

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

   12. Applied rewrites 80.9%

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

   if 1.2400000000000001e147 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

       7. times-frac N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       8. pow2 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-/.f64 61.1

           \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333 \]

   11. Applied rewrites 61.1%

       \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.061:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\_m\right)}{\cos k} \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.061)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma
        (fma
         (fma 0.03432539682539683 (* k k) 0.08611111111111111)
         (* k k)
         0.16666666666666666)
        (* k k)
        1.0)
       (* k k))))
    (/
     2.0
     (*
      (/ (* k (* k t_m)) (cos k))
      (/ (- 0.5 (* 0.5 (cos (* 2.0 k)))) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.061) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (((k * (k * t_m)) / cos(k)) * ((0.5 - (0.5 * cos((2.0 * k)))) / (l * l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.061)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t_m)) / cos(k)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 0.061], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.060999999999999999

   1. Initial program 37.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 0.060999999999999999 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

Alternative 9: 73.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t\_m} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.55)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma
        (fma
         (fma 0.03432539682539683 (* k k) 0.08611111111111111)
         (* k k)
         0.16666666666666666)
        (* k k)
        1.0)
       (* k k))))
    (* (/ (pow (/ l k) 2.0) t_m) -0.3333333333333333))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.55) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = (pow((l / k), 2.0) / t_m) * -0.3333333333333333;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.55)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t_m) * -0.3333333333333333);
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 1.55], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004

   1. Initial program 37.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 77.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 1.55000000000000004 < k

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{3} \]

       7. times-frac N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{3} \]

       8. pow2 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-/.f64 58.3

           \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333 \]

   11. Applied rewrites 58.3%

       \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \color{blue}{-0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\_m\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.1e+164)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma
        (fma
         (fma 0.03432539682539683 (* k k) 0.08611111111111111)
         (* k k)
         0.16666666666666666)
        (* k k)
        1.0)
       (* k k))))
    (/
     2.0
     (* (/ (* k (* k t_m)) (fma -0.5 (* k k) 1.0)) (/ (* k k) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.1e+164) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(fma(0.03432539682539683, (k * k), 0.08611111111111111), (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (((k * (k * t_m)) / fma(-0.5, (k * k), 1.0)) * ((k * k) / (l * l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 2.1e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(fma(0.03432539682539683, Float64(k * k), 0.08611111111111111), Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t_m)) / fma(-0.5, Float64(k * k), 1.0)) * Float64(Float64(k * k) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 2.1e+164], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.03432539682539683 * N[(k * k), $MachinePrecision] + 0.08611111111111111), $MachinePrecision] * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.0999999999999999e164

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. pow2 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 95.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + {k}^{2} \cdot \left(\frac{31}{360} + \frac{173}{5040} \cdot {k}^{2}\right)\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

   10. Applied rewrites 74.6%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, k \cdot k, 0.08611111111111111\right), k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.0999999999999999e164 < l

   1. Initial program 27.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.9

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

   7. Applied rewrites 58.9%

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

   8. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 58.5

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

   10. Applied rewrites 58.5%

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

   11. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 58.1

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

   13. Applied rewrites 58.1%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\_m\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.1e+164)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (*
       (fma (fma 0.08611111111111111 (* k k) 0.16666666666666666) (* k k) 1.0)
       (* k k))))
    (/
     2.0
     (* (/ (* k (* k t_m)) (fma -0.5 (* k k) 1.0)) (/ (* k k) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.1e+164) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(fma(0.08611111111111111, (k * k), 0.16666666666666666), (k * k), 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (((k * (k * t_m)) / fma(-0.5, (k * k), 1.0)) * ((k * k) / (l * l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 2.1e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(fma(0.08611111111111111, Float64(k * k), 0.16666666666666666), Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t_m)) / fma(-0.5, Float64(k * k), 1.0)) * Float64(Float64(k * k) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 2.1e+164], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08611111111111111 * N[(k * k), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.0999999999999999e164

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 80.2

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 80.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      9. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   7. Applied rewrites 95.0%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + {k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right)\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) + 1\right) \cdot {k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}\right) \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{31}{360} \cdot {k}^{2}, {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{31}{360} \cdot {k}^{2} + \frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, {k}^{2}, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot {k}^{2}\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{31}{360}, k \cdot k, \frac{1}{6}\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \]

      13. lift-*.f64 74.9

          \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \]

   10. Applied rewrites 74.9%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08611111111111111, k \cdot k, 0.16666666666666666\right), k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.0999999999999999e164 < l

   1. Initial program 27.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 58.5

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\ell \cdot \ell}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      5. lower-*.f64 58.9

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

      2. lift-*.f64 58.5

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

   10. Applied rewrites 58.5%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\color{blue}{\ell} \cdot \ell}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{1 + \frac{-1}{2} \cdot {k}^{2}} \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\frac{-1}{2} \cdot {k}^{2} + 1} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(\frac{-1}{2}, {k}^{2}, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(\frac{-1}{2}, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       4. lift-*.f64 58.1

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

   13. Applied rewrites 58.1%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\_m\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.1e+164)
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (* (fma 0.16666666666666666 (* k k) 1.0) (* k k))))
    (/
     2.0
     (* (/ (* k (* k t_m)) (fma -0.5 (* k k) 1.0)) (/ (* k k) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.1e+164) {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * (fma(0.16666666666666666, (k * k), 1.0) * (k * k)));
	} else {
		tmp = 2.0 / (((k * (k * t_m)) / fma(-0.5, (k * k), 1.0)) * ((k * k) / (l * l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 2.1e+164)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(fma(0.16666666666666666, Float64(k * k), 1.0) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t_m)) / fma(-0.5, Float64(k * k), 1.0)) * Float64(Float64(k * k) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 2.1e+164], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.0999999999999999e164

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 80.2

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 80.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

      9. frac-times N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      14. times-frac N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   7. Applied rewrites 95.0%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(1 + \frac{1}{6} \cdot {k}^{2}\right) \cdot {k}^{\color{blue}{2}}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot {k}^{2}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right) \cdot {k}^{2}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot {k}^{2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot {k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \]

      8. lift-*.f64 74.8

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \]

   10. Applied rewrites 74.8%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.0999999999999999e164 < l

   1. Initial program 27.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

      3. times-frac N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

      14. lift-*.f64 58.5

          \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. Applied rewrites 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\ell \cdot \ell}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

      5. lower-*.f64 58.9

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

      2. lift-*.f64 58.5

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

   10. Applied rewrites 58.5%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\color{blue}{\ell} \cdot \ell}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{1 + \frac{-1}{2} \cdot {k}^{2}} \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\frac{-1}{2} \cdot {k}^{2} + 1} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(\frac{-1}{2}, {k}^{2}, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(\frac{-1}{2}, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

       4. lift-*.f64 58.1

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

   13. Applied rewrites 58.1%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\mathsf{fma}\left(-0.5, k \cdot k, 1\right)} \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(k \cdot k\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (/ k l) (/ (* k t_m) l)) (* k k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k / l) * ((k * t_m) / l)) * (k * k)));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((k / l) * ((k * t_m) / l)) * (k * k)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k / l) * ((k * t_m) / l)) * (k * k)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((k / l) * ((k * t_m) / l)) * (k * k)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(k * k))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((k / l) * ((k * t_m) / l)) * (k * k)));
end

Wolfram

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_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 78.0

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 78.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

   9. frac-times N/A

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

   14. times-frac N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

7. Applied rewrites 93.2%

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 71.3

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)} \]

10. Applied rewrites 71.3%

    \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]
11. Add Preprocessing

Alternative 14: 73.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(k \cdot t\_m\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* k t_m) k) (* (/ k l) (/ k l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k * t_m) * k) * ((k / l) * (k / l))));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((k * t_m) * k) * ((k / l) * (k / l))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k * t_m) * k) * ((k / l) * (k / l))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((k * t_m) * k) * ((k / l) * (k / l))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(k * t_m) * k) * Float64(Float64(k / l) * Float64(k / l)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((k * t_m) * k) * ((k / l) * (k / l))));
end

Wolfram

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_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 78.0

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 78.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \ell}} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{\color{blue}{2}}}} \]

   9. frac-times N/A

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos \color{blue}{k} \cdot {\ell}^{2}}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

   14. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{\cos k} \cdot {\ell}^{2}}} \]

   15. times-frac N/A

       \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}} \]

7. Applied rewrites 93.1%

   \[\leadsto \frac{2}{\frac{{\sin k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{\frac{k}{\ell}} \cdot \frac{k}{\ell}\right)} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(\frac{k}{\color{blue}{\ell}} \cdot \frac{k}{\ell}\right)} \]

   5. lift-*.f64 69.4

      \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \]

10. Applied rewrites 69.4%

    \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(\color{blue}{\frac{k}{\ell}} \cdot \frac{k}{\ell}\right)} \]
11. Add Preprocessing

Alternative 15: 65.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(k \cdot t\_m\right) \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* k t_m) k) (/ (* k k) (* l l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k * t_m) * k) * ((k * k) / (l * l))));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((k * t_m) * k) * ((k * k) / (l * l))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((k * t_m) * k) * ((k * k) / (l * l))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((k * t_m) * k) * ((k * k) / (l * l))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(k * t_m) * k) * Float64(Float64(k * k) / Float64(l * l)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((k * t_m) * k) * ((k * k) / (l * l))));
end

Wolfram

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_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\cos k \cdot \color{blue}{{\ell}^{2}}}} \]

   3. times-frac N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{{\ell}^{2}}} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{{\ell}^{2}}}} \]

   11. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\color{blue}{\ell}}^{2}}} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}} \]

   13. pow2 N/A

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

   14. lift-*.f64 78.0

       \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\ell}}} \]

5. Applied rewrites 78.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin \color{blue}{k}}^{2}}{\ell \cdot \ell}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

   5. lower-*.f64 81.8

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}} \]

7. Applied rewrites 81.8%

   \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \ell}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{{k}^{2}}{\color{blue}{\ell} \cdot \ell}} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

   2. lift-*.f64 68.2

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

10. Applied rewrites 68.2%

    \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\cos k} \cdot \frac{k \cdot k}{\color{blue}{\ell} \cdot \ell}} \]

11. Taylor expanded in k around 0

    \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]
12. Step-by-step derivation

    1. pow2 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

    2. associate-*r* N/A

       \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]

    3. *-commutative N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{k \cdot \color{blue}{k}}{\ell \cdot \ell}} \]

    5. lift-*.f64 64.9

       \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}} \]

13. Applied rewrites 64.9%

    \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]
14. Add Preprocessing

Alternative 16: 61.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 (* (* k k) (* k k))) (/ (* l l) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / ((k * k) * (k * k))) * ((l * l) / t_m));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 / ((k * k) * (k * k))) * ((l * l) / t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / ((k * k) * (k * k))) * ((l * l) / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 / ((k * k) * (k * k))) * ((l * l) / t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l * l) / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 / ((k * k) * (k * k))) * ((l * l) / t_m));
end

Wolfram

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_, k_] := N[(t$95$s * N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. times-frac N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{{\ell}^{2}}}{t} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{\color{blue}{2}}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.2

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t} \]

5. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2}{{k}^{\left(2 + 2\right)}} \cdot \frac{\ell \cdot \ell}{t} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot {k}^{2}} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   5. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]

   7. pow2 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \]

   8. lift-*.f64 60.2

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 60.2%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]
8. Add Preprocessing

Alternative 17: 20.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t\_m} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* -0.11666666666666667 (* l l)) t_m)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((-0.11666666666666667 * (l * l)) / t_m);
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((-0.11666666666666667d0) * (l * l)) / t_m)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((-0.11666666666666667 * (l * l)) / t_m);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((-0.11666666666666667 * (l * l)) / t_m)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(-0.11666666666666667 * Float64(l * l)) / t_m))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((-0.11666666666666667 * (l * l)) / t_m);
end

Wolfram

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_, k_] := N[(t$95$s * N[(N[(-0.11666666666666667 * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 22.3

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 22.3%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{t} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot \left(\ell \cdot \ell\right)}{t} \]

   9. lift-*.f64 22.3

      \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]

10. Applied rewrites 22.3%

    \[\leadsto \frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t} \]
11. Add Preprocessing

Alternative 18: 20.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* -0.11666666666666667 (/ (* l l) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (-0.11666666666666667 * ((l * l) / t_m));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((-0.11666666666666667d0) * ((l * l) / t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (-0.11666666666666667 * ((l * l) / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (-0.11666666666666667 * ((l * l) / t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(-0.11666666666666667 * Float64(Float64(l * l) / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (-0.11666666666666667 * ((l * l) / t_m));
end

Wolfram

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_, k_] := N[(t$95$s * N[(-0.11666666666666667 * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 22.3

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 22.3%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Add Preprocessing

Alternative 19: 18.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t\_m}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* -0.11666666666666667 (* l (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (-0.11666666666666667 * (l * (l / t_m)));
}

Fortran

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, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((-0.11666666666666667d0) * (l * (l / t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (-0.11666666666666667 * (l * (l / t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (-0.11666666666666667 * (l * (l / t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(-0.11666666666666667 * Float64(l * Float64(l / t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (-0.11666666666666667 * (l * (l / t_m)));
end

Wolfram

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_, k_] := N[(t$95$s * N[(-0.11666666666666667 * N[(l * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{{\ell}^{2}}{\color{blue}{t}} \]

   2. pow2 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   4. lift-*.f64 22.3

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

8. Applied rewrites 22.3%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \frac{\ell \cdot \ell}{t} \]

   3. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]

   5. lower-/.f64 20.5

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

10. Applied rewrites 20.5%

    \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025073 
(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))))