Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 93.9%

Time: 9.2s

Alternatives: 20

Speedup: 5.0×

• 40.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.9% 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: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-54}:\\ \;\;\;\;t\_2 \cdot \frac{\ell + \ell}{k \cdot \left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\frac{1}{t\_m}}{\sin k} \cdot \frac{2}{\sin k}\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (* (cos k) l) k)))
   (*
    t_s
    (if (<= t_m 1.25e-54)
      (* t_2 (/ (+ l l) (* k (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m))))
      (* (* t_2 (/ l k)) (* (/ (/ 1.0 t_m) (sin k)) (/ 2.0 (sin k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (cos(k) * l) / k;
	double tmp;
	if (t_m <= 1.25e-54) {
		tmp = t_2 * ((l + l) / (k * ((0.5 - (cos((k + k)) * 0.5)) * t_m)));
	} else {
		tmp = (t_2 * (l / k)) * (((1.0 / t_m) / sin(k)) * (2.0 / sin(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) :: t_2
    real(8) :: tmp
    t_2 = (cos(k) * l) / k
    if (t_m <= 1.25d-54) then
        tmp = t_2 * ((l + l) / (k * ((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m)))
    else
        tmp = (t_2 * (l / k)) * (((1.0d0 / t_m) / sin(k)) * (2.0d0 / sin(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 t_2 = (Math.cos(k) * l) / k;
	double tmp;
	if (t_m <= 1.25e-54) {
		tmp = t_2 * ((l + l) / (k * ((0.5 - (Math.cos((k + k)) * 0.5)) * t_m)));
	} else {
		tmp = (t_2 * (l / k)) * (((1.0 / t_m) / Math.sin(k)) * (2.0 / Math.sin(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):
	t_2 = (math.cos(k) * l) / k
	tmp = 0
	if t_m <= 1.25e-54:
		tmp = t_2 * ((l + l) / (k * ((0.5 - (math.cos((k + k)) * 0.5)) * t_m)))
	else:
		tmp = (t_2 * (l / k)) * (((1.0 / t_m) / math.sin(k)) * (2.0 / math.sin(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(cos(k) * l) / k)
	tmp = 0.0
	if (t_m <= 1.25e-54)
		tmp = Float64(t_2 * Float64(Float64(l + l) / Float64(k * Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m))));
	else
		tmp = Float64(Float64(t_2 * Float64(l / k)) * Float64(Float64(Float64(1.0 / t_m) / sin(k)) * Float64(2.0 / sin(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)
	t_2 = (cos(k) * l) / k;
	tmp = 0.0;
	if (t_m <= 1.25e-54)
		tmp = t_2 * ((l + l) / (k * ((0.5 - (cos((k + k)) * 0.5)) * t_m)));
	else
		tmp = (t_2 * (l / k)) * (((1.0 / t_m) / sin(k)) * (2.0 / sin(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_] := Block[{t$95$2 = N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.25e-54], N[(t$95$2 * N[(N[(l + l), $MachinePrecision] / N[(k * N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000004e-54

   1. Initial program 37.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 69.4%

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

   6. Applied rewrites 67.0%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 84.1

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

   8. Applied rewrites 84.1%

      \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{2}}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 91.2%

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

   if 1.25000000000000004e-54 < t

   1. Initial program 34.7%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.2%

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

   6. Applied rewrites 69.0%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 81.7

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

   8. Applied rewrites 81.7%

      \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{2}}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. sqr-sin-a-rev N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 95.9%

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

Alternative 2: 92.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k \cdot \ell}{k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-54}:\\ \;\;\;\;t\_2 \cdot \frac{\ell + \ell}{k \cdot \left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\ell}{k}\right) \cdot \frac{2}{{\sin k}^{2} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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) k)))
   (*
    t_s
    (if (<= t_m 1.25e-54)
      (* t_2 (/ (+ l l) (* k (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m))))
      (* (* t_2 (/ l k)) (/ 2.0 (* (pow (sin k) 2.0) 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) {
	double t_2 = (cos(k) * l) / k;
	double tmp;
	if (t_m <= 1.25e-54) {
		tmp = t_2 * ((l + l) / (k * ((0.5 - (cos((k + k)) * 0.5)) * t_m)));
	} else {
		tmp = (t_2 * (l / k)) * (2.0 / (pow(sin(k), 2.0) * t_m));
	}
	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 = (cos(k) * l) / k
    if (t_m <= 1.25d-54) then
        tmp = t_2 * ((l + l) / (k * ((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m)))
    else
        tmp = (t_2 * (l / k)) * (2.0d0 / ((sin(k) ** 2.0d0) * t_m))
    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) / k;
	double tmp;
	if (t_m <= 1.25e-54) {
		tmp = t_2 * ((l + l) / (k * ((0.5 - (Math.cos((k + k)) * 0.5)) * t_m)));
	} else {
		tmp = (t_2 * (l / k)) * (2.0 / (Math.pow(Math.sin(k), 2.0) * t_m));
	}
	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) / k
	tmp = 0
	if t_m <= 1.25e-54:
		tmp = t_2 * ((l + l) / (k * ((0.5 - (math.cos((k + k)) * 0.5)) * t_m)))
	else:
		tmp = (t_2 * (l / k)) * (2.0 / (math.pow(math.sin(k), 2.0) * t_m))
	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(cos(k) * l) / k)
	tmp = 0.0
	if (t_m <= 1.25e-54)
		tmp = Float64(t_2 * Float64(Float64(l + l) / Float64(k * Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m))));
	else
		tmp = Float64(Float64(t_2 * Float64(l / k)) * Float64(2.0 / Float64((sin(k) ^ 2.0) * t_m)));
	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) / k;
	tmp = 0.0;
	if (t_m <= 1.25e-54)
		tmp = t_2 * ((l + l) / (k * ((0.5 - (cos((k + k)) * 0.5)) * t_m)));
	else
		tmp = (t_2 * (l / k)) * (2.0 / ((sin(k) ^ 2.0) * t_m));
	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[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.25e-54], N[(t$95$2 * N[(N[(l + l), $MachinePrecision] / N[(k * N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000004e-54

   1. Initial program 37.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 69.4%

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

   6. Applied rewrites 67.0%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 84.1

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

   8. Applied rewrites 84.1%

      \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{2}}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 91.2%

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

   if 1.25000000000000004e-54 < t

   1. Initial program 34.7%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.2%

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

   6. Applied rewrites 69.0%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 81.7

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

   8. Applied rewrites 81.7%

      \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{2}}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. unpow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 93.4

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

   10. Applied rewrites 93.4%

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

Alternative 3: 83.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.000125:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.000125)
    (* (* (/ 2.0 k) (/ l (* k t_m))) (/ l (* k k)))
    (*
     (* (/ (* (cos k) l) k) (/ l k))
     (/ 2.0 (* (- 0.5 (* (cos (+ k k)) 0.5)) 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) {
	double tmp;
	if (k <= 0.000125) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = (((cos(k) * l) / k) * (l / k)) * (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t_m));
	}
	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 <= 0.000125d0) then
        tmp = ((2.0d0 / k) * (l / (k * t_m))) * (l / (k * k))
    else
        tmp = (((cos(k) * l) / k) * (l / k)) * (2.0d0 / ((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m))
    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 <= 0.000125) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = (((Math.cos(k) * l) / k) * (l / k)) * (2.0 / ((0.5 - (Math.cos((k + k)) * 0.5)) * t_m));
	}
	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 <= 0.000125:
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k))
	else:
		tmp = (((math.cos(k) * l) / k) * (l / k)) * (2.0 / ((0.5 - (math.cos((k + k)) * 0.5)) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.000125)
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) / k) * Float64(l / k)) * Float64(2.0 / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m)));
	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 <= 0.000125)
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	else
		tmp = (((cos(k) * l) / k) * (l / k)) * (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t_m));
	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, 0.000125], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-4

   1. Initial program 37.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.9

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

   6. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      5. pow2 N/A

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

      6. associate-/l/ N/A

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

      7. count-2-rev N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 80.7

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

   10. Applied rewrites 80.7%

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

   if 1.25e-4 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 72.4%

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

   6. Applied rewrites 71.5%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 90.9

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

   8. Applied rewrites 90.9%

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

Alternative 4: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.000125:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.000125)
    (* (* (/ 2.0 k) (/ l (* k t_m))) (/ l (* k k)))
    (*
     (* (* (cos k) (/ l k)) (/ l k))
     (/ 2.0 (* (- 0.5 (* (cos (+ k k)) 0.5)) 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) {
	double tmp;
	if (k <= 0.000125) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = ((cos(k) * (l / k)) * (l / k)) * (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t_m));
	}
	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 <= 0.000125d0) then
        tmp = ((2.0d0 / k) * (l / (k * t_m))) * (l / (k * k))
    else
        tmp = ((cos(k) * (l / k)) * (l / k)) * (2.0d0 / ((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m))
    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 <= 0.000125) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = ((Math.cos(k) * (l / k)) * (l / k)) * (2.0 / ((0.5 - (Math.cos((k + k)) * 0.5)) * t_m));
	}
	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 <= 0.000125:
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k))
	else:
		tmp = ((math.cos(k) * (l / k)) * (l / k)) * (2.0 / ((0.5 - (math.cos((k + k)) * 0.5)) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.000125)
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(cos(k) * Float64(l / k)) * Float64(l / k)) * Float64(2.0 / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m)));
	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 <= 0.000125)
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	else
		tmp = ((cos(k) * (l / k)) * (l / k)) * (2.0 / ((0.5 - (cos((k + k)) * 0.5)) * t_m));
	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, 0.000125], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-4

   1. Initial program 37.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.9

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

   6. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      5. pow2 N/A

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

      6. associate-/l/ N/A

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

      7. count-2-rev N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 80.7

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

   10. Applied rewrites 80.7%

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

   if 1.25e-4 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 72.4%

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

   6. Applied rewrites 71.5%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 90.9

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

   8. Applied rewrites 90.9%

      \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{2}}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-/.f64 90.8

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

   10. Applied rewrites 90.8%

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

Alternative 5: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.000102:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{k \cdot \left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.000102)
    (* (* (/ 2.0 k) (/ l (* k t_m))) (/ l (* k k)))
    (*
     (/ (* (cos k) l) k)
     (/ (+ l l) (* k (* (- 0.5 (* (cos (+ k k)) 0.5)) 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) {
	double tmp;
	if (k <= 0.000102) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = ((cos(k) * l) / k) * ((l + l) / (k * ((0.5 - (cos((k + k)) * 0.5)) * t_m)));
	}
	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 <= 0.000102d0) then
        tmp = ((2.0d0 / k) * (l / (k * t_m))) * (l / (k * k))
    else
        tmp = ((cos(k) * l) / k) * ((l + l) / (k * ((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m)))
    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 <= 0.000102) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = ((Math.cos(k) * l) / k) * ((l + l) / (k * ((0.5 - (Math.cos((k + k)) * 0.5)) * t_m)));
	}
	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 <= 0.000102:
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k))
	else:
		tmp = ((math.cos(k) * l) / k) * ((l + l) / (k * ((0.5 - (math.cos((k + k)) * 0.5)) * t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.000102)
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(cos(k) * l) / k) * Float64(Float64(l + l) / Float64(k * Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m))));
	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 <= 0.000102)
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	else
		tmp = ((cos(k) * l) / k) * ((l + l) / (k * ((0.5 - (cos((k + k)) * 0.5)) * t_m)));
	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, 0.000102], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(k * N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999999e-4

   1. Initial program 37.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.9

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

   6. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      5. pow2 N/A

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

      6. associate-/l/ N/A

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

      7. count-2-rev N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 80.7

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

   10. Applied rewrites 80.7%

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

   if 1.01999999999999999e-4 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 72.4%

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

   6. Applied rewrites 71.5%

      \[\leadsto \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{2}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 90.8

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

   8. Applied rewrites 90.8%

      \[\leadsto \left(\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{2}}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   10. Applied rewrites 93.5%

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

Alternative 6: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.000125:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell + \ell\right) \cdot \ell\right) \cdot \cos k}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

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 0.000125)
    (* (* (/ 2.0 k) (/ l (* k t_m))) (/ l (* k k)))
    (/
     (* (* (+ l l) l) (cos k))
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) 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) {
	double tmp;
	if (k <= 0.000125) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = (((l + l) * l) * cos(k)) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 0.000125d0) then
        tmp = ((2.0d0 / k) * (l / (k * t_m))) * (l / (k * k))
    else
        tmp = (((l + l) * l) * cos(k)) / ((((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * k) * 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 <= 0.000125) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = (((l + l) * l) * Math.cos(k)) / ((((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 0.000125:
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k))
	else:
		tmp = (((l + l) * l) * math.cos(k)) / ((((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 0.000125)
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) * l) * cos(k)) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * 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 <= 0.000125)
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	else
		tmp = (((l + l) * l) * cos(k)) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * 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, 0.000125], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-4

   1. Initial program 37.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.9

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

   6. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      5. pow2 N/A

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

      6. associate-/l/ N/A

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

      7. count-2-rev N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 80.7

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

   10. Applied rewrites 80.7%

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

   if 1.25e-4 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 72.4%

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 77.4%

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

Alternative 7: 75.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (* (* (/ 2.0 k) (/ l (* k t_m))) (/ l (* k k)))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = ((2.0d0 / k) * (l / (k * t_m))) * (l / (k * k))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = ((2.0 / k) * (l / (k * t_m))) * (l / (k * k));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{t} \cdot \frac{\ell}{k \cdot k} \]

      5. pow2 N/A

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

      6. associate-/l/ N/A

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

      7. count-2-rev N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 80.3

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

   10. Applied rewrites 80.3%

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

   if 6.2e6 < k

   1. Initial program 31.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

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

Alternative 8: 75.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k}}{t\_m} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (* (/ (/ (+ l l) (* k k)) t_m) (/ l (* k k)))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = (((l + l) / (k * k)) / t_m) * (l / (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = (((l + l) / (k * k)) / t_m) * (l / (k * k))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = (((l + l) / (k * k)) / t_m) * (l / (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = (((l + l) / (k * k)) / t_m) * (l / (k * k))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k * k)) / t_m) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = (((l + l) / (k * k)) / t_m) * (l / (k * k));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

   if 6.2e6 < k

   1. Initial program 31.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

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

Alternative 9: 74.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (* (/ (/ (+ l l) (* k k)) k) (/ l (* k t_m)))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = (((l + l) / (k * k)) / k) * (l / (k * t_m));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = (((l + l) / (k * k)) / k) * (l / (k * t_m))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = (((l + l) / (k * k)) / k) * (l / (k * t_m));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = (((l + l) / (k * k)) / k) * (l / (k * t_m))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k * k)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = (((l + l) / (k * k)) / k) * (l / (k * t_m));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. count-2-rev N/A

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{k} \cdot \frac{\ell}{k \cdot t} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k}}{k} \cdot \frac{\ell}{k \cdot t} \]

      17. lower-/.f64 N/A

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

      18. lower-*.f64 79.6

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

   8. Applied rewrites 79.6%

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

   if 6.2e6 < k

   1. Initial program 31.8%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (* (/ (* 2.0 l) (* k k)) (/ l (* k (* k t_m))))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t_m)));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = ((2.0d0 * l) / (k * k)) * (l / (k * (k * t_m)))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t_m)));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t_m)))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t_m))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t_m)));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      11. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}} \]

      19. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      20. lift-*.f64 79.5

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   6. Applied rewrites 79.5%

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

      5. lower-*.f64 79.5

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

   8. Applied rewrites 79.5%

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

   if 6.2e6 < k

   1. Initial program 31.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 55.7

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{\left(k \cdot k\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (/ (* (/ (+ l l) (* k k)) l) (* (* k k) t_m))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = (((l + l) / (k * k)) * l) / ((k * k) * t_m);
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = (((l + l) / (k * k)) * l) / ((k * k) * t_m)
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = (((l + l) / (k * k)) * l) / ((k * k) * t_m);
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = (((l + l) / (k * k)) * l) / ((k * k) * t_m)
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(Float64(l + l) / Float64(k * k)) * l) / Float64(Float64(k * k) * t_m));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = (((l + l) / (k * k)) * l) / ((k * k) * t_m);
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      11. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}} \]

      19. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      20. lift-*.f64 79.5

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   6. Applied rewrites 79.5%

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{\left(k \cdot k\right) \cdot t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{{k}^{2} \cdot t} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{\color{blue}{{k}^{2} \cdot t}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{\color{blue}{{k}^{2}} \cdot t} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{{k}^{2} \cdot t} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \ell}{k \cdot k} \cdot \ell}{{\color{blue}{k}}^{2} \cdot t} \]

      14. count-2-rev N/A

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{{k}^{2} \cdot t} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{{k}^{2} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{\left(k \cdot k\right) \cdot t} \]

      18. lift-*.f64 77.8

          \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   8. Applied rewrites 77.8%

      \[\leadsto \frac{\frac{\ell + \ell}{k \cdot k} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   if 6.2e6 < k

   1. Initial program 31.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 55.7

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (* (/ (+ l l) k) (/ l (* k (* (* k k) t_m))))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = ((l + l) / k) * (l / (k * ((k * k) * t_m)));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = ((l + l) / k) * (l / (k * ((k * k) * t_m)))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = ((l + l) / k) * (l / (k * ((k * k) * t_m)));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = ((l + l) / k) * (l / (k * ((k * k) * t_m)))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(l + l) / k) * Float64(l / Float64(k * Float64(Float64(k * k) * t_m))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = ((l + l) / k) * (l / (k * ((k * k) * t_m)));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      11. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}} \]

      19. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      20. lift-*.f64 79.5

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   6. Applied rewrites 79.5%

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{\left(k \cdot k\right) \cdot t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      7. frac-times N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}} \]

      10. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      13. count-2-rev N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      18. pow2 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left({k}^{2} \cdot t\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      20. pow2 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      22. lift-*.f64 78.2

          \[\leadsto \frac{\ell + \ell}{k} \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

   8. Applied rewrites 78.2%

      \[\leadsto \frac{\ell + \ell}{k} \cdot \color{blue}{\frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

   if 6.2e6 < k

   1. Initial program 31.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 55.7

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{\ell + \ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot k} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (* (/ (+ l l) (* (* (* k k) k) k)) (/ l t_m))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = ((l + l) / (((k * k) * k) * k)) * (l / t_m);
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = ((l + l) / (((k * k) * k) * k)) * (l / t_m)
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = ((l + l) / (((k * k) * k) * k)) * (l / t_m);
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = ((l + l) / (((k * k) * k) * k)) * (l / t_m)
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(l + l) / Float64(Float64(Float64(k * k) * k) * k)) * Float64(l / t_m));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = ((l + l) / (((k * k) * k) * k)) * (l / t_m);
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      11. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}} \]

      19. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      20. lift-*.f64 79.5

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   6. Applied rewrites 79.5%

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{\left(k \cdot k\right) \cdot t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      7. frac-times N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot \color{blue}{t}} \]

      12. pow-prod-up N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{\left(2 + 2\right)} \cdot t} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{4} \cdot t} \]

      14. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{4}} \cdot \color{blue}{\frac{\ell}{t}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{4}} \cdot \color{blue}{\frac{\ell}{t}} \]

   8. Applied rewrites 73.5%

      \[\leadsto \frac{\ell + \ell}{\left(\left(k \cdot k\right) \cdot k\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t}} \]

   if 6.2e6 < k

   1. Initial program 31.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 55.7

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6200000:\\ \;\;\;\;\frac{\left(\ell + \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 6200000.0)
    (/ (* (+ l l) l) (* (* (* k k) t_m) (* k k)))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 6200000.0) {
		tmp = ((l + l) * l) / (((k * k) * t_m) * (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0d0) then
        tmp = ((l + l) * l) / (((k * k) * t_m) * (k * k))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 6200000.0) {
		tmp = ((l + l) * l) / (((k * k) * t_m) * (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 6200000.0:
		tmp = ((l + l) * l) / (((k * k) * t_m) * (k * k))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 6200000.0)
		tmp = Float64(Float64(Float64(l + l) * l) / Float64(Float64(Float64(k * k) * t_m) * Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 6200000.0)
		tmp = ((l + l) * l) / (((k * k) * t_m) * (k * k));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 6200000.0], N[(N[(N[(l + l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e6

   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. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\left(2 + 2\right)} \cdot t} \]

      8. pow-prod-up N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot {k}^{2}\right) \cdot t} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{\left(k \cdot k\right)}^{2} \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot t} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]

      11. times-frac N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{{k}^{2} \cdot t} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      15. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{t}} \]

      19. pow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      20. lift-*.f64 79.5

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

   6. Applied rewrites 79.5%

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell}}{\left(k \cdot k\right) \cdot t} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{{k}^{2}} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      7. frac-times N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left({k}^{2} \cdot {k}^{2}\right) \cdot \color{blue}{t}} \]

      12. pow-prod-up N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{\left(2 + 2\right)} \cdot t} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{4} \cdot t} \]

      14. associate-*r* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{4}} \cdot t} \]

      15. pow2 N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{\color{blue}{4}} \cdot t} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      17. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{{k}^{4}} \cdot t} \]

      20. count-2-rev N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{{\color{blue}{k}}^{4} \cdot t} \]

      21. lower-+.f64 N/A

          \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{{\color{blue}{k}}^{4} \cdot t} \]

   8. Applied rewrites 68.3%

      \[\leadsto \frac{\left(\ell + \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   if 6.2e6 < k

   1. Initial program 31.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 55.7

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.7%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.6

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.6%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 52:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(0.5 - 0.5\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 52.0)
    (/ (* 2.0 (* l l)) (* (* (- 0.5 0.5) t_m) (* k k)))
    (* (/ l k) (/ (* l -0.3333333333333333) (* k 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) {
	double tmp;
	if (k <= 52.0) {
		tmp = (2.0 * (l * l)) / (((0.5 - 0.5) * t_m) * (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 52.0d0) then
        tmp = (2.0d0 * (l * l)) / (((0.5d0 - 0.5d0) * t_m) * (k * k))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t_m))
    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 <= 52.0) {
		tmp = (2.0 * (l * l)) / (((0.5 - 0.5) * t_m) * (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	}
	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 <= 52.0:
		tmp = (2.0 * (l * l)) / (((0.5 - 0.5) * t_m) * (k * k))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m))
	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 <= 52.0)
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(Float64(0.5 - 0.5) * t_m) * Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t_m)));
	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 <= 52.0)
		tmp = (2.0 * (l * l)) / (((0.5 - 0.5) * t_m) * (k * k));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t_m));
	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, 52.0], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 52

   1. Initial program 37.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot \color{blue}{t}\right) \cdot \left(k \cdot k\right)} \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 61.5

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   7. Applied rewrites 61.5%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot \color{blue}{t}\right) \cdot \left(k \cdot k\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   if 52 < k

   1. Initial program 31.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      9. lift-*.f64 55.2

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 55.2%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

      13. lower-*.f64 58.0

          \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

   9. Applied rewrites 58.0%

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 30.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t\_m}\right) \end{array} \]

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 (* (/ l k) (/ (* l -0.3333333333333333) (* k 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 * ((l / k) * ((l * -0.3333333333333333) / (k * 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 * ((l / k) * ((l * (-0.3333333333333333d0)) / (k * 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 * ((l / k) * ((l * -0.3333333333333333) / (k * 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 * ((l / k) * ((l * -0.3333333333333333) / (k * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * 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 * ((l / k) * ((l * -0.3333333333333333) / (k * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 29.2

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.2%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{\left(k \cdot k\right) \cdot t} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \left(\ell \cdot \frac{-1}{3}\right)}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

   8. times-frac N/A

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k} \cdot t} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot \color{blue}{t}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{k \cdot t} \]

   13. lower-*.f64 30.7

       \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \]

9. Applied rewrites 30.7%

   \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]
10. Add Preprocessing

Alternative 17: 29.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t\_m}\right) \end{array} \]

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 (* (/ -0.3333333333333333 k) (/ (* l l) (* k 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 * ((-0.3333333333333333 / k) * ((l * l) / (k * 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 * (((-0.3333333333333333d0) / k) * ((l * l) / (k * 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 * ((-0.3333333333333333 / k) * ((l * l) / (k * 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 * ((-0.3333333333333333 / k) * ((l * l) / (k * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(-0.3333333333333333 / k) * Float64(Float64(l * l) / Float64(k * 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 * ((-0.3333333333333333 / k) * ((l * l) / (k * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(-0.3333333333333333 / k), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 29.2

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.2%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

   9. times-frac N/A

      \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{{\ell}^{2}}{\color{blue}{k \cdot t}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{{\ell}^{2}}{\color{blue}{k \cdot t}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{{\ell}^{2}}{\color{blue}{k} \cdot t} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{{\ell}^{2}}{k \cdot \color{blue}{t}} \]

   13. pow2 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \]

   15. lower-*.f64 29.8

       \[\leadsto \frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \]

9. Applied rewrites 29.8%

   \[\leadsto \frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot t}} \]
10. Add Preprocessing

Alternative 18: 29.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\_m\right)} \end{array} \]

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 (/ (* (* l l) -0.3333333333333333) (* k (* k 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 * (((l * l) * -0.3333333333333333) / (k * (k * 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 * (((l * l) * (-0.3333333333333333d0)) / (k * (k * 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 * (((l * l) * -0.3333333333333333) / (k * (k * 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 * (((l * l) * -0.3333333333333333) / (k * (k * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(k * Float64(k * 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 * (((l * l) * -0.3333333333333333) / (k * (k * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 29.2

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.2%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]

   5. lower-*.f64 29.6

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

9. Applied rewrites 29.6%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]
10. Add Preprocessing

Alternative 19: 29.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t\_m} \cdot -0.3333333333333333\right) \end{array} \]

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 (* (/ (* l l) (* (* k k) t_m)) -0.3333333333333333)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * l) / ((k * k) * t_m)) * -0.3333333333333333);
}

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 * (((l * l) / ((k * k) * t_m)) * (-0.3333333333333333d0))
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 * (((l * l) / ((k * k) * t_m)) * -0.3333333333333333);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l * l) / ((k * k) * t_m)) * -0.3333333333333333)

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(Float64(l * l) / Float64(Float64(k * k) * t_m)) * -0.3333333333333333))
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 * (((l * l) / ((k * k) * t_m)) * -0.3333333333333333);
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[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 29.2

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.2%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

   17. lift-*.f64 29.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

9. Applied rewrites 29.2%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
10. Add Preprocessing

Alternative 20: 29.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\left(k \cdot k\right) \cdot t\_m}\right) \end{array} \]

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 (* (* l l) (/ -0.3333333333333333 (* (* k k) 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 * ((l * l) * (-0.3333333333333333 / ((k * k) * 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 * ((l * l) * ((-0.3333333333333333d0) / ((k * k) * 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 * ((l * l) * (-0.3333333333333333 / ((k * k) * 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 * ((l * l) * (-0.3333333333333333 / ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(Float64(k * k) * 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 * ((l * l) * (-0.3333333333333333 / ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(k \cdot \ell\right) \cdot \left(k \cdot \ell\right), 2 \cdot \left(\ell \cdot \ell\right)\right)}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   9. lift-*.f64 29.2

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.2%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   5. associate-/l* N/A

      \[\leadsto {\ell}^{2} \cdot \frac{\frac{-1}{3}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   6. lower-*.f64 N/A

      \[\leadsto {\ell}^{2} \cdot \frac{\frac{-1}{3}}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   7. pow2 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   11. pow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{{k}^{2} \cdot t} \]

   12. lower-/.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{{k}^{2} \cdot \color{blue}{t}} \]

   13. pow2 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{-1}{3}}{\left(k \cdot k\right) \cdot t} \]

   15. lift-*.f64 29.2

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

9. Applied rewrites 29.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(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))))