Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.7% → 94.8%

Time: 8.7s

Alternatives: 21

Speedup: 8.6×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.7% 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: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{\ell} \cdot \left(\frac{k}{\cos k} \cdot \frac{k \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot t\_m}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

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 6e+100)
      (/ 2.0 (* (/ t_2 l) (* (/ k (cos k)) (/ (* k t_m) l))))
      (/ 2.0 (* (/ (* t_2 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 t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 6e+100) {
		tmp = 2.0 / ((t_2 / l) * ((k / cos(k)) * ((k * t_m) / l)));
	} else {
		tmp = 2.0 / (((t_2 * t_m) / cos(k)) * ((k / l) * (k / l)));
	}
	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 <= 6d+100) then
        tmp = 2.0d0 / ((t_2 / l) * ((k / cos(k)) * ((k * t_m) / l)))
    else
        tmp = 2.0d0 / (((t_2 * t_m) / cos(k)) * ((k / l) * (k / l)))
    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 <= 6e+100) {
		tmp = 2.0 / ((t_2 / l) * ((k / Math.cos(k)) * ((k * t_m) / l)));
	} else {
		tmp = 2.0 / (((t_2 * t_m) / Math.cos(k)) * ((k / l) * (k / l)));
	}
	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 <= 6e+100:
		tmp = 2.0 / ((t_2 / l) * ((k / math.cos(k)) * ((k * t_m) / l)))
	else:
		tmp = 2.0 / (((t_2 * t_m) / math.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)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 6e+100)
		tmp = Float64(2.0 / Float64(Float64(t_2 / l) * Float64(Float64(k / cos(k)) * Float64(Float64(k * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) / cos(k)) * Float64(Float64(k / l) * Float64(k / l))));
	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 <= 6e+100)
		tmp = 2.0 / ((t_2 / l) * ((k / cos(k)) * ((k * t_m) / l)));
	else
		tmp = 2.0 / (((t_2 * t_m) / cos(k)) * ((k / l) * (k / l)));
	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, 6e+100], N[(2.0 / N[(N[(t$95$2 / l), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * 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 t < 5.99999999999999971e100

   1. Initial program 48.2%

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

   2. Taylor expanded in t around 0

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

      1. 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 75.5

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

   4. Applied rewrites 75.5%

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

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

   6. Applied rewrites 96.4%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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. frac-times N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. associate-/l* N/A

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

   8. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. associate-*r/ N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 N/A

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

   10. Applied rewrites 96.3%

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

   if 5.99999999999999971e100 < t

   1. Initial program 13.9%

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

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. 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 74.3

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

   4. Applied rewrites 74.3%

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

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

   6. Applied rewrites 91.8%

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

Alternative 2: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\cos k \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot t\_m}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

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 2e+98)
      (/ 2.0 (* t_2 (* (/ k l) (/ (* k t_m) (* (cos k) l)))))
      (/ 2.0 (* (/ (* t_2 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 t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 2e+98) {
		tmp = 2.0 / (t_2 * ((k / l) * ((k * t_m) / (cos(k) * l))));
	} else {
		tmp = 2.0 / (((t_2 * t_m) / cos(k)) * ((k / l) * (k / l)));
	}
	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 <= 2d+98) then
        tmp = 2.0d0 / (t_2 * ((k / l) * ((k * t_m) / (cos(k) * l))))
    else
        tmp = 2.0d0 / (((t_2 * t_m) / cos(k)) * ((k / l) * (k / l)))
    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 <= 2e+98) {
		tmp = 2.0 / (t_2 * ((k / l) * ((k * t_m) / (Math.cos(k) * l))));
	} else {
		tmp = 2.0 / (((t_2 * t_m) / Math.cos(k)) * ((k / l) * (k / l)));
	}
	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 <= 2e+98:
		tmp = 2.0 / (t_2 * ((k / l) * ((k * t_m) / (math.cos(k) * l))))
	else:
		tmp = 2.0 / (((t_2 * t_m) / math.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)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 2e+98)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k / l) * Float64(Float64(k * t_m) / Float64(cos(k) * l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) / cos(k)) * Float64(Float64(k / l) * Float64(k / l))));
	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 <= 2e+98)
		tmp = 2.0 / (t_2 * ((k / l) * ((k * t_m) / (cos(k) * l))));
	else
		tmp = 2.0 / (((t_2 * t_m) / cos(k)) * ((k / l) * (k / l)));
	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, 2e+98], N[(2.0 / N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * 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 t < 2e98

   1. Initial program 48.2%

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

   2. Taylor expanded in t around 0

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

      1. 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 75.5

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

   4. Applied rewrites 75.5%

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

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

   6. Applied rewrites 96.5%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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. frac-times N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. associate-/l* N/A

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

   8. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

   if 2e98 < t

   1. Initial program 14.4%

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

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. 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 74.3

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

   4. Applied rewrites 74.3%

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

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

   6. Applied rewrites 91.7%

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

Alternative 3: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k \cdot \ell\\ t_3 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{t\_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_3 \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\_m\right)}{t\_2}}\\ \end{array} \end{array} \end{array} \]

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 (* (cos k) l)) (t_3 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 5.8e+25)
      (/ 2.0 (* t_3 (* (/ k l) (/ (* k t_m) t_2))))
      (/ 2.0 (/ (* t_3 (* (* (/ k l) k) t_m)) t_2))))))

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 = cos(k) * l;
	double t_3 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 5.8e+25) {
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)));
	} else {
		tmp = 2.0 / ((t_3 * (((k / l) * k) * t_m)) / t_2);
	}
	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) :: t_3
    real(8) :: tmp
    t_2 = cos(k) * l
    t_3 = sin(k) ** 2.0d0
    if (t_m <= 5.8d+25) then
        tmp = 2.0d0 / (t_3 * ((k / l) * ((k * t_m) / t_2)))
    else
        tmp = 2.0d0 / ((t_3 * (((k / l) * k) * t_m)) / t_2)
    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.cos(k) * l;
	double t_3 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 5.8e+25) {
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)));
	} else {
		tmp = 2.0 / ((t_3 * (((k / l) * k) * t_m)) / t_2);
	}
	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.cos(k) * l
	t_3 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 5.8e+25:
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)))
	else:
		tmp = 2.0 / ((t_3 * (((k / l) * k) * t_m)) / t_2)
	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(cos(k) * l)
	t_3 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 5.8e+25)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(k / l) * Float64(Float64(k * t_m) / t_2))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_3 * Float64(Float64(Float64(k / l) * k) * t_m)) / t_2));
	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 = cos(k) * l;
	t_3 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 5.8e+25)
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)));
	else
		tmp = 2.0 / ((t_3 * (((k / l) * k) * t_m)) / t_2);
	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[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e+25], N[(2.0 / N[(t$95$3 * N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$3 * N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999998e25

   1. Initial program 45.7%

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

   2. Taylor expanded in t around 0

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

      1. 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 75.0

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

   4. Applied rewrites 75.0%

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

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

   6. Applied rewrites 97.3%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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. frac-times N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. associate-/l* N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 97.3

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

   10. Applied rewrites 97.3%

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

   if 5.7999999999999998e25 < t

   1. Initial program 26.1%

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

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. 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 75.3

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

   4. Applied rewrites 75.3%

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

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

   6. Applied rewrites 87.1%

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

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

   8. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 90.8

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

   10. Applied rewrites 90.8%

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

Alternative 4: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k \cdot \ell\\ t_3 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{t\_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{t\_3 \cdot t\_m}{\ell}}{t\_2}}\\ \end{array} \end{array} \end{array} \]

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 (* (cos k) l)) (t_3 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 3.2e+81)
      (/ 2.0 (* t_3 (* (/ k l) (/ (* k t_m) t_2))))
      (/ 2.0 (/ (* (* k k) (/ (* t_3 t_m) l)) t_2))))))

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 = cos(k) * l;
	double t_3 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 3.2e+81) {
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)));
	} else {
		tmp = 2.0 / (((k * k) * ((t_3 * t_m) / l)) / t_2);
	}
	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) :: t_3
    real(8) :: tmp
    t_2 = cos(k) * l
    t_3 = sin(k) ** 2.0d0
    if (t_m <= 3.2d+81) then
        tmp = 2.0d0 / (t_3 * ((k / l) * ((k * t_m) / t_2)))
    else
        tmp = 2.0d0 / (((k * k) * ((t_3 * t_m) / l)) / t_2)
    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.cos(k) * l;
	double t_3 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 3.2e+81) {
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)));
	} else {
		tmp = 2.0 / (((k * k) * ((t_3 * t_m) / l)) / t_2);
	}
	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.cos(k) * l
	t_3 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 3.2e+81:
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)))
	else:
		tmp = 2.0 / (((k * k) * ((t_3 * t_m) / l)) / t_2)
	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(cos(k) * l)
	t_3 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 3.2e+81)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(k / l) * Float64(Float64(k * t_m) / t_2))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(t_3 * t_m) / l)) / t_2));
	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 = cos(k) * l;
	t_3 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 3.2e+81)
		tmp = 2.0 / (t_3 * ((k / l) * ((k * t_m) / t_2)));
	else
		tmp = 2.0 / (((k * k) * ((t_3 * t_m) / l)) / t_2);
	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[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.2e+81], N[(2.0 / N[(t$95$3 * N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$3 * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2e81

   1. Initial program 47.8%

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

   2. Taylor expanded in t around 0

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

      1. 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 75.4

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

   4. Applied rewrites 75.4%

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

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

   6. Applied rewrites 96.7%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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. frac-times N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. associate-/l* N/A

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

   8. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-*.f64 96.7

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

   10. Applied rewrites 96.7%

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

   if 3.2e81 < t

   1. Initial program 17.5%

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

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. 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 74.6

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

   4. Applied rewrites 74.6%

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

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

   6. Applied rewrites 85.5%

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

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

   8. Applied rewrites 87.5%

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

   9. Taylor expanded in t around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-sin.f64 N/A

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

       9. lift-pow.f64 89.2

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

   11. Applied rewrites 89.2%

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

Alternative 5: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\cos k \cdot \ell}\right)} \end{array} \]

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 (* (pow (sin k) 2.0) (* (/ k l) (/ (* k t_m) (* (cos 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 / (pow(sin(k), 2.0) * ((k / l) * ((k * t_m) / (cos(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 / ((sin(k) ** 2.0d0) * ((k / l) * ((k * t_m) / (cos(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 / (Math.pow(Math.sin(k), 2.0) * ((k / l) * ((k * t_m) / (Math.cos(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 / (math.pow(math.sin(k), 2.0) * ((k / l) * ((k * t_m) / (math.cos(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((sin(k) ^ 2.0) * Float64(Float64(k / l) * Float64(Float64(k * t_m) / Float64(cos(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 / ((sin(k) ^ 2.0) * ((k / l) * ((k * t_m) / (cos(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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

2. Taylor expanded in t around 0

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

   1. 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 75.1

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

4. Applied rewrites 75.1%

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

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

6. Applied rewrites 92.6%

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
7. 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. frac-times N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. times-frac N/A

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

   17. pow2 N/A

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

   18. associate-/l* N/A

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

8. Applied rewrites 75.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lift-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-*.f64 92.6

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

10. Applied rewrites 92.6%

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

Alternative 6: 84.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}{\cos k}}\\ \end{array} \end{array} \]

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 2.35e-7)
    (/ 2.0 (/ (* (pow (sin k) 2.0) (* (/ k l) (* k t_m))) l))
    (/
     2.0
     (*
      (* (/ k l) (/ (* k t_m) l))
      (/ (- 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 tmp;
	if (k <= 2.35e-7) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((k / l) * (k * t_m))) / l);
	} else {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * ((0.5 - (0.5 * cos((2.0 * k)))) / cos(k)));
	}
	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) :: tmp
    if (k <= 2.35d-7) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((k / l) * (k * t_m))) / l)
    else
        tmp = 2.0d0 / (((k / l) * ((k * t_m) / l)) * ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) / cos(k)))
    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 tmp;
	if (k <= 2.35e-7) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((k / l) * (k * t_m))) / l);
	} else {
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * ((0.5 - (0.5 * Math.cos((2.0 * k)))) / Math.cos(k)));
	}
	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):
	tmp = 0
	if k <= 2.35e-7:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((k / l) * (k * t_m))) / l)
	else:
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * ((0.5 - (0.5 * math.cos((2.0 * k)))) / math.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)
	tmp = 0.0
	if (k <= 2.35e-7)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) * Float64(k * t_m))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(k * t_m) / l)) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) / cos(k))));
	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)
	tmp = 0.0;
	if (k <= 2.35e-7)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((k / l) * (k * t_m))) / l);
	else
		tmp = 2.0 / (((k / l) * ((k * t_m) / l)) * ((0.5 - (0.5 * cos((2.0 * k)))) / cos(k)));
	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_] := N[(t$95$s * If[LessEqual[k, 2.35e-7], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * 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 < 2.35e-7

   1. Initial program 38.6%

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

   2. Taylor expanded in t around 0

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

      1. 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 76.3

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

   4. Applied rewrites 76.3%

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

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

   6. Applied rewrites 92.3%

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

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

   8. Applied rewrites 92.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 81.0%

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

   if 2.35e-7 < k

   1. Initial program 31.3%

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

   2. Taylor expanded in t around 0

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

      1. 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 71.7

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

   4. Applied rewrites 71.7%

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

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

   6. Applied rewrites 93.6%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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 93.1

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

   8. Applied rewrites 93.1%

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

Alternative 7: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{\ell} \cdot \left(k \cdot t\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot t\_2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\_2}{\cos k \cdot \ell}}\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    t_s
    (if (<= k 2.35e-7)
      (/ 2.0 (/ (* (pow (sin k) 2.0) t_2) l))
      (/ 2.0 (/ (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t_2) (* (cos 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 t_2 = (k / l) * (k * t_m);
	double tmp;
	if (k <= 2.35e-7) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * t_2) / l);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k)))) * t_2) / (cos(k) * l));
	}
	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 = (k / l) * (k * t_m)
    if (k <= 2.35d-7) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * t_2) / l)
    else
        tmp = 2.0d0 / (((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * t_2) / (cos(k) * l))
    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 = (k / l) * (k * t_m);
	double tmp;
	if (k <= 2.35e-7) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * t_2) / l);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * Math.cos((2.0 * k)))) * t_2) / (Math.cos(k) * l));
	}
	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 = (k / l) * (k * t_m)
	tmp = 0
	if k <= 2.35e-7:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * t_2) / l)
	else:
		tmp = 2.0 / (((0.5 - (0.5 * math.cos((2.0 * k)))) * t_2) / (math.cos(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)
	t_2 = Float64(Float64(k / l) * Float64(k * t_m))
	tmp = 0.0
	if (k <= 2.35e-7)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * t_2) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t_2) / Float64(cos(k) * l)));
	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 = (k / l) * (k * t_m);
	tmp = 0.0;
	if (k <= 2.35e-7)
		tmp = 2.0 / (((sin(k) ^ 2.0) * t_2) / l);
	else
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k)))) * t_2) / (cos(k) * l));
	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[(k / l), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.35e-7], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.35e-7

   1. Initial program 38.6%

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

   2. Taylor expanded in t around 0

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

      1. 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 76.3

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

   4. Applied rewrites 76.3%

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

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

   6. Applied rewrites 92.3%

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

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

   8. Applied rewrites 92.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 81.0%

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

   if 2.35e-7 < k

   1. Initial program 31.3%

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

   2. Taylor expanded in t around 0

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

      1. 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 71.7

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

   4. Applied rewrites 71.7%

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

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

   6. Applied rewrites 93.6%

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

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

   8. Applied rewrites 92.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 91.6

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

   10. Applied rewrites 91.6%

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

Alternative 8: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}{\cos k \cdot \left(\ell \cdot \ell\right)}}\\ \end{array} \end{array} \]

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 2.35e-7)
    (/ 2.0 (/ (* (pow (sin k) 2.0) (* (/ k l) (* k t_m))) l))
    (/
     2.0
     (/
      (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) (* (* k k) t_m))
      (* (cos 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 <= 2.35e-7) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((k / l) * (k * t_m))) / l);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k)))) * ((k * k) * t_m)) / (cos(k) * (l * l)));
	}
	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) :: tmp
    if (k <= 2.35d-7) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((k / l) * (k * t_m))) / l)
    else
        tmp = 2.0d0 / (((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * ((k * k) * t_m)) / (cos(k) * (l * l)))
    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 tmp;
	if (k <= 2.35e-7) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((k / l) * (k * t_m))) / l);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * Math.cos((2.0 * k)))) * ((k * k) * t_m)) / (Math.cos(k) * (l * l)));
	}
	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):
	tmp = 0
	if k <= 2.35e-7:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((k / l) * (k * t_m))) / l)
	else:
		tmp = 2.0 / (((0.5 - (0.5 * math.cos((2.0 * k)))) * ((k * k) * t_m)) / (math.cos(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 <= 2.35e-7)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) * Float64(k * t_m))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * Float64(Float64(k * k) * t_m)) / Float64(cos(k) * Float64(l * l))));
	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)
	tmp = 0.0;
	if (k <= 2.35e-7)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((k / l) * (k * t_m))) / l);
	else
		tmp = 2.0 / (((0.5 - (0.5 * cos((2.0 * k)))) * ((k * k) * t_m)) / (cos(k) * (l * l)));
	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_] := N[(t$95$s * If[LessEqual[k, 2.35e-7], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.35e-7

   1. Initial program 38.6%

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

   2. Taylor expanded in t around 0

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

      1. 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 76.3

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

   4. Applied rewrites 76.3%

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

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

   6. Applied rewrites 92.3%

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

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

   8. Applied rewrites 92.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 81.0%

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

   if 2.35e-7 < k

   1. Initial program 31.3%

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

   2. Taylor expanded in t around 0

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

      1. 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 71.7

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

   4. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell \cdot \ell}}} \]
   5. 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-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 71.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

Alternative 9: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot t\_m\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \frac{\left(k \cdot k\right) \cdot t\_m}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

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 2.35e-7)
    (/ 2.0 (/ (* (pow (sin k) 2.0) (* (/ k l) (* k t_m))) l))
    (/
     2.0
     (*
      (- 0.5 (* 0.5 (cos (* 2.0 k))))
      (/ (* (* k k) t_m) (* (* (cos 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 <= 2.35e-7) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((k / l) * (k * t_m))) / l);
	} else {
		tmp = 2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * (((k * k) * t_m) / ((cos(k) * l) * l)));
	}
	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) :: tmp
    if (k <= 2.35d-7) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((k / l) * (k * t_m))) / l)
    else
        tmp = 2.0d0 / ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * (((k * k) * t_m) / ((cos(k) * l) * l)))
    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 tmp;
	if (k <= 2.35e-7) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((k / l) * (k * t_m))) / l);
	} else {
		tmp = 2.0 / ((0.5 - (0.5 * Math.cos((2.0 * k)))) * (((k * k) * t_m) / ((Math.cos(k) * l) * l)));
	}
	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):
	tmp = 0
	if k <= 2.35e-7:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((k / l) * (k * t_m))) / l)
	else:
		tmp = 2.0 / ((0.5 - (0.5 * math.cos((2.0 * k)))) * (((k * k) * t_m) / ((math.cos(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 <= 2.35e-7)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) * Float64(k * t_m))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * Float64(Float64(Float64(k * k) * t_m) / Float64(Float64(cos(k) * l) * l))));
	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)
	tmp = 0.0;
	if (k <= 2.35e-7)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((k / l) * (k * t_m))) / l);
	else
		tmp = 2.0 / ((0.5 - (0.5 * cos((2.0 * k)))) * (((k * k) * t_m) / ((cos(k) * l) * l)));
	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_] := N[(t$95$s * If[LessEqual[k, 2.35e-7], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.35e-7

   1. Initial program 38.6%

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

   2. Taylor expanded in t around 0

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

      1. 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 76.3

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

   4. Applied rewrites 76.3%

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

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

   6. Applied rewrites 92.3%

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

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

   8. Applied rewrites 92.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 81.0%

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

   if 2.35e-7 < k

   1. Initial program 31.3%

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

   2. Taylor expanded in t around 0

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

      1. 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 71.7

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

   4. Applied rewrites 71.7%

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

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

   6. Applied rewrites 93.6%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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. frac-times N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. associate-/l* N/A

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

   8. Applied rewrites 71.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 71.5

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

   10. Applied rewrites 71.5%

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

Alternative 10: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.1e-95)
    (/ 2.0 (* (pow (sin k) 2.0) (* (/ (* k k) l) (/ t_m l))))
    (/ 2.0 (* (/ (* (* k 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 (t_m <= 1.1e-95) {
		tmp = 2.0 / (pow(sin(k), 2.0) * (((k * k) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / cos(k)) * ((k / l) * (k / l)));
	}
	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) :: tmp
    if (t_m <= 1.1d-95) then
        tmp = 2.0d0 / ((sin(k) ** 2.0d0) * (((k * k) / l) * (t_m / l)))
    else
        tmp = 2.0d0 / ((((k * k) * t_m) / cos(k)) * ((k / l) * (k / l)))
    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 tmp;
	if (t_m <= 1.1e-95) {
		tmp = 2.0 / (Math.pow(Math.sin(k), 2.0) * (((k * k) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / Math.cos(k)) * ((k / l) * (k / l)));
	}
	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):
	tmp = 0
	if t_m <= 1.1e-95:
		tmp = 2.0 / (math.pow(math.sin(k), 2.0) * (((k * k) / l) * (t_m / l)))
	else:
		tmp = 2.0 / ((((k * k) * t_m) / math.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 (t_m <= 1.1e-95)
		tmp = Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(Float64(Float64(k * k) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / cos(k)) * Float64(Float64(k / l) * Float64(k / l))));
	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)
	tmp = 0.0;
	if (t_m <= 1.1e-95)
		tmp = 2.0 / ((sin(k) ^ 2.0) * (((k * k) / l) * (t_m / l)));
	else
		tmp = 2.0 / ((((k * k) * t_m) / cos(k)) * ((k / l) * (k / l)));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-95], N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $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 t < 1.0999999999999999e-95

   1. Initial program 32.0%

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

   2. Taylor expanded in t around 0

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

      1. 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 73.5

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

   4. Applied rewrites 73.5%

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

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

   6. Applied rewrites 97.3%

      \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}} \]
   7. 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. frac-times N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. associate-/l* N/A

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. times-frac N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-/.f64 71.4

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

   11. Applied rewrites 71.4%

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

   if 1.0999999999999999e-95 < t

   1. Initial program 39.2%

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

   2. Taylor expanded in t around 0

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

      1. 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 75.9

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 11: 64.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-67}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{t\_m}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= k 3.8e-67)
    (*
     (* (/ l k) (/ l k))
     (/ (/ (fma (* k k) -0.3333333333333333 2.0) t_m) (* k k)))
    (/ 2.0 (* (/ (* (* k k) t_m) (cos 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) {
	double tmp;
	if (k <= 3.8e-67) {
		tmp = ((l / k) * (l / k)) * ((fma((k * k), -0.3333333333333333, 2.0) / t_m) / (k * k));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / cos(k)) * ((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 (k <= 3.8e-67)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(fma(Float64(k * k), -0.3333333333333333, 2.0) / t_m) / Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / cos(k)) * 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[k, 3.8e-67], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.79999999999999988e-67

   1. Initial program 39.9%

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

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. 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 75.8

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 53.9%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, \left(\ell \cdot \ell\right) \cdot 2\right)}{t}}{\color{blue}{{k}^{4}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{-1}{3}, {\left(\ell \cdot k\right)}^{2}, \left(\ell \cdot \ell\right) \cdot 2\right)}{t}}{{\color{blue}{k}}^{4}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + \left(\ell \cdot \ell\right) \cdot 2}{t}}{{k}^{4}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + \left(\ell \cdot \ell\right) \cdot 2}{t}}{{k}^{4}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + \left(\ell \cdot \ell\right) \cdot 2}{t}}{{k}^{4}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + \left(\ell \cdot \ell\right) \cdot 2}{t}}{{k}^{4}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\left(\ell \cdot k\right)}^{2} + \left(\ell \cdot \ell\right) \cdot 2}{t}}{{k}^{4}} \]

      8. lift-pow.f64 N/A

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

   9. Applied rewrites 65.8%

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

   if 3.79999999999999988e-67 < k

   1. Initial program 29.6%

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

   2. Taylor expanded in t around 0

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

      1. 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 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 61.8

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

   7. Applied rewrites 61.8%

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

Alternative 12: 73.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \end{array} \]

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 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) {
	return t_s * (2.0 / ((((k * k) * t_m) / cos(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 * k) * t_m) / cos(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 * k) * t_m) / Math.cos(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 * k) * t_m) / math.cos(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(Float64(k * k) * t_m) / cos(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 * k) * t_m) / cos(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[(N[(k * k), $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. Initial program 36.7%

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

2. Taylor expanded in t around 0

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

   1. 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 75.1

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

4. Applied rewrites 75.1%

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

5. Taylor expanded in k around 0

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

   1. pow2 N/A

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

   2. pow2 N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 73.6

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

7. Applied rewrites 73.6%

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

Alternative 13: 74.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}{\ell \cdot \cos k}} \end{array} \]

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) l) (* (* k k) t_m)) (* l (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) {
	return t_s * (2.0 / ((((k * k) / l) * ((k * k) * t_m)) / (l * cos(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 * k) / l) * ((k * k) * t_m)) / (l * cos(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 * k) / l) * ((k * k) * t_m)) / (l * Math.cos(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 * k) / l) * ((k * k) * t_m)) / (l * math.cos(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(Float64(k * k) / l) * Float64(Float64(k * k) * t_m)) / Float64(l * cos(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 * k) / l) * ((k * k) * t_m)) / (l * cos(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[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

2. Taylor expanded in t around 0

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

   1. 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 75.1

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

4. Applied rewrites 75.1%

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

5. Taylor expanded in k around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 67.7

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

7. Applied rewrites 67.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. pow2 N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

9. Applied rewrites 74.1%

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

Alternative 14: 72.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(k \cdot k\right)\right)} \end{array} \]

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
  (/
   2.0
   (*
    (* (/ k l) (/ (* k t_m) l))
    (* (fma 0.16666666666666666 (* k k) 1.0) (* 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)) * (fma(0.16666666666666666, (k * k), 1.0) * (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(fma(0.16666666666666666, Float64(k * k), 1.0) * Float64(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[(N[(0.16666666666666666 * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

2. Taylor expanded in t around 0

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

   1. 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 75.1

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

4. Applied rewrites 75.1%

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

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

6. Applied rewrites 92.6%

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

7. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left({k}^{2} \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {k}^{2}\right)}\right)} \]
8. 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 72.8

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

9. Applied rewrites 72.8%

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
10. Add Preprocessing

Alternative 15: 67.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t\_m} \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

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 (<= t_m 1.7e-127)
    (/ (* (* (/ l t_m) l) 2.0) (* (* k k) (* k k)))
    (/ 2.0 (* (* (* k k) t_m) (/ (* 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 (t_m <= 1.7e-127) {
		tmp = (((l / t_m) * l) * 2.0) / ((k * k) * (k * k));
	} else {
		tmp = 2.0 / (((k * k) * t_m) * ((k * k) / (l * l)));
	}
	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) :: tmp
    if (t_m <= 1.7d-127) then
        tmp = (((l / t_m) * l) * 2.0d0) / ((k * k) * (k * k))
    else
        tmp = 2.0d0 / (((k * k) * t_m) * ((k * k) / (l * l)))
    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 tmp;
	if (t_m <= 1.7e-127) {
		tmp = (((l / t_m) * l) * 2.0) / ((k * k) * (k * k));
	} else {
		tmp = 2.0 / (((k * k) * t_m) * ((k * k) / (l * l)));
	}
	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):
	tmp = 0
	if t_m <= 1.7e-127:
		tmp = (((l / t_m) * l) * 2.0) / ((k * k) * (k * k))
	else:
		tmp = 2.0 / (((k * k) * t_m) * ((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 (t_m <= 1.7e-127)
		tmp = Float64(Float64(Float64(Float64(l / t_m) * l) * 2.0) / Float64(Float64(k * k) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(k * k) / Float64(l * l))));
	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)
	tmp = 0.0;
	if (t_m <= 1.7e-127)
		tmp = (((l / t_m) * l) * 2.0) / ((k * k) * (k * k));
	else
		tmp = 2.0 / (((k * k) * t_m) * ((k * k) / (l * l)));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-127], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6999999999999999e-127

   1. Initial program 29.4%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{\color{blue}{4}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. lift-*.f64 21.2

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   6. Applied rewrites 21.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 64.9

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

   9. Applied rewrites 64.9%

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

   if 1.6999999999999999e-127 < t

   1. Initial program 39.7%

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

   2. Taylor expanded in t around 0

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

      1. 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 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 69.8

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}} \]
   9. 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. lift-*.f64 N/A

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

      3. lift-*.f64 68.7

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

   10. Applied rewrites 68.7%

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

Alternative 16: 65.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{t\_m} \cdot \ell\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{t\_2 \cdot 2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k \cdot k} \cdot t\_2\\ \end{array} \end{array} \end{array} \]

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 (* (/ l t_m) l)))
   (*
    t_s
    (if (<= k 1.05e+68)
      (/ (* t_2 2.0) (* (* k k) (* k k)))
      (* (/ -0.3333333333333333 (* k k)) t_2)))))

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 = (l / t_m) * l;
	double tmp;
	if (k <= 1.05e+68) {
		tmp = (t_2 * 2.0) / ((k * k) * (k * k));
	} else {
		tmp = (-0.3333333333333333 / (k * k)) * t_2;
	}
	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 = (l / t_m) * l
    if (k <= 1.05d+68) then
        tmp = (t_2 * 2.0d0) / ((k * k) * (k * k))
    else
        tmp = ((-0.3333333333333333d0) / (k * k)) * t_2
    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 = (l / t_m) * l;
	double tmp;
	if (k <= 1.05e+68) {
		tmp = (t_2 * 2.0) / ((k * k) * (k * k));
	} else {
		tmp = (-0.3333333333333333 / (k * k)) * t_2;
	}
	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 = (l / t_m) * l
	tmp = 0
	if k <= 1.05e+68:
		tmp = (t_2 * 2.0) / ((k * k) * (k * k))
	else:
		tmp = (-0.3333333333333333 / (k * k)) * t_2
	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(l / t_m) * l)
	tmp = 0.0
	if (k <= 1.05e+68)
		tmp = Float64(Float64(t_2 * 2.0) / Float64(Float64(k * k) * Float64(k * k)));
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(k * k)) * t_2);
	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 = (l / t_m) * l;
	tmp = 0.0;
	if (k <= 1.05e+68)
		tmp = (t_2 * 2.0) / ((k * k) * (k * k));
	else
		tmp = (-0.3333333333333333 / (k * k)) * t_2;
	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[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.05e+68], N[(N[(t$95$2 * 2.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e68

   1. Initial program 37.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{\color{blue}{4}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{\left(2 + \color{blue}{2}\right)}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{2} \cdot \color{blue}{{k}^{2}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot {\color{blue}{k}}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot \frac{7}{120}, \frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. lift-*.f64 31.6

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   6. Applied rewrites 31.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 66.8

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

   9. Applied rewrites 66.8%

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

   if 1.05e68 < k

   1. Initial program 34.3%

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

   2. Taylor expanded in t around 0

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

      1. 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 67.5

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

   4. Applied rewrites 67.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 36.2%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 59.6

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

   10. Applied rewrites 59.6%

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

Alternative 17: 62.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k \cdot k} \cdot \left(\frac{\ell}{t\_m} \cdot \ell\right)\\ \end{array} \end{array} \]

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 1.05e+68)
    (* (/ 2.0 (* (* k k) (* k k))) (/ (* l l) t_m))
    (* (/ -0.3333333333333333 (* k k)) (* (/ l t_m) 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 <= 1.05e+68) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	} else {
		tmp = (-0.3333333333333333 / (k * k)) * ((l / t_m) * l);
	}
	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) :: tmp
    if (k <= 1.05d+68) then
        tmp = (2.0d0 / ((k * k) * (k * k))) * ((l * l) / t_m)
    else
        tmp = ((-0.3333333333333333d0) / (k * k)) * ((l / t_m) * l)
    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 tmp;
	if (k <= 1.05e+68) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	} else {
		tmp = (-0.3333333333333333 / (k * k)) * ((l / t_m) * l);
	}
	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):
	tmp = 0
	if k <= 1.05e+68:
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m)
	else:
		tmp = (-0.3333333333333333 / (k * k)) * ((l / t_m) * 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 <= 1.05e+68)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l * l) / t_m));
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(k * k)) * Float64(Float64(l / t_m) * l));
	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)
	tmp = 0.0;
	if (k <= 1.05e+68)
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	else
		tmp = (-0.3333333333333333 / (k * k)) * ((l / t_m) * l);
	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_] := N[(t$95$s * If[LessEqual[k, 1.05e+68], 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], N[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e68

   1. Initial program 37.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   3. 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 62.8

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

   4. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   5. 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 62.8

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

   6. Applied rewrites 62.8%

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

   if 1.05e68 < k

   1. Initial program 34.3%

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

   2. Taylor expanded in t around 0

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

      1. 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 67.5

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

   4. Applied rewrites 67.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 36.2%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 59.6

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

   10. Applied rewrites 59.6%

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

Alternative 18: 73.4% 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 36.7%

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

2. Taylor expanded in t around 0

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

   1. 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 75.1

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

4. Applied rewrites 75.1%

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

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

6. Applied rewrites 92.6%

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

7. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {k}^{\color{blue}{2}}} \]
8. 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 73.4

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

9. Applied rewrites 73.4%

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

Alternative 19: 30.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{-0.3333333333333333}{k \cdot k} \cdot \left(\frac{\ell}{t\_m} \cdot \ell\right)\right) \end{array} \]

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 (* (/ -0.3333333333333333 (* k k)) (* (/ l t_m) 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 * ((-0.3333333333333333 / (k * k)) * ((l / t_m) * 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 * (((-0.3333333333333333d0) / (k * k)) * ((l / t_m) * 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 * ((-0.3333333333333333 / (k * k)) * ((l / t_m) * 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 * ((-0.3333333333333333 / (k * k)) * ((l / t_m) * 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(Float64(-0.3333333333333333 / Float64(k * k)) * Float64(Float64(l / t_m) * 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 * ((-0.3333333333333333 / (k * k)) * ((l / t_m) * 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[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

2. Taylor expanded in t around 0

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

   1. 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 75.1

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

4. Applied rewrites 75.1%

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

5. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 49.7%

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

8. Taylor expanded in k around inf

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

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. associate-*r/ N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 30.4

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

10. Applied rewrites 30.4%

    \[\leadsto \frac{-0.3333333333333333}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \ell\right)} \]
11. Add Preprocessing

Alternative 20: 20.3% 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 36.7%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 27.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. 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 20.3

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 20.3%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
8. Add Preprocessing

Alternative 21: 17.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t\_m}\right)\right) \end{array} \]

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 36.7%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 27.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(k \cdot k\right), \frac{\ell \cdot \ell}{t} \cdot 0.058333333333333334, \frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333\right), k \cdot k, \frac{\ell \cdot \ell}{t} \cdot 2\right)}{{k}^{4}}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
6. 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 20.3

      \[\leadsto -0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

7. Applied rewrites 20.3%

   \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]
8. 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 17.8

      \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

9. Applied rewrites 17.8%

   \[\leadsto -0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{t}}\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025095 
(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))))