Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 95.2%

Time: 8.0s

Alternatives: 14

Speedup: 5.7×

• 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: 35.4% 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: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-161}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\_m\right) \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\sin k \cdot t\_m} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-161)
    (*
     2.0
     (* l (* (/ (cos k) k) (/ l (* (* (fma (cos (+ k k)) -0.5 0.5) t_m) k)))))
    (* (/ 2.0 (tan k)) (* (/ (* (/ l k) (/ l k)) (* (sin k) t_m)) 1.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.2e-161) {
		tmp = 2.0 * (l * ((cos(k) / k) * (l / ((fma(cos((k + k)), -0.5, 0.5) * t_m) * k))));
	} else {
		tmp = (2.0 / tan(k)) * ((((l / k) * (l / k)) / (sin(k) * t_m)) * 1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.2e-161)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(cos(k) / k) * Float64(l / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * t_m) * k)))));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(sin(k) * t_m)) * 1.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-161], N[(2.0 * N[(l * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.20000000000000036e-161

   1. Initial program 35.4%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 83.0

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

   8. Applied rewrites 83.0%

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

   if 7.20000000000000036e-161 < t

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 2: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\cos k \cdot \ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\_m\right) \cdot k} \cdot \frac{\ell + \ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\sin k \cdot t\_m} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-161)
    (*
     (/ (* (cos k) l) (* (* (fma (cos (+ k k)) -0.5 0.5) t_m) k))
     (/ (+ l l) k))
    (* (/ 2.0 (tan k)) (* (/ (* (/ l k) (/ l k)) (* (sin k) t_m)) 1.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.2e-161) {
		tmp = ((cos(k) * l) / ((fma(cos((k + k)), -0.5, 0.5) * t_m) * k)) * ((l + l) / k);
	} else {
		tmp = (2.0 / tan(k)) * ((((l / k) * (l / k)) / (sin(k) * t_m)) * 1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.2e-161)
		tmp = Float64(Float64(Float64(cos(k) * l) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * t_m) * k)) * Float64(Float64(l + l) / k));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(sin(k) * t_m)) * 1.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-161], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.20000000000000036e-161

   1. Initial program 35.4%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

   8. Applied rewrites 83.8%

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

   if 7.20000000000000036e-161 < t

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 3: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\sin k \cdot t\_m} \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-161)
    (*
     (/ (* (cos k) l) k)
     (/ (+ l l) (* (* (fma (cos (+ k k)) -0.5 0.5) t_m) k)))
    (* (/ 2.0 (tan k)) (* (/ (* (/ l k) (/ l k)) (* (sin k) t_m)) 1.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.2e-161) {
		tmp = ((cos(k) * l) / k) * ((l + l) / ((fma(cos((k + k)), -0.5, 0.5) * t_m) * k));
	} else {
		tmp = (2.0 / tan(k)) * ((((l / k) * (l / k)) / (sin(k) * t_m)) * 1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.2e-161)
		tmp = Float64(Float64(Float64(cos(k) * l) / k) * Float64(Float64(l + l) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * t_m) * k)));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(sin(k) * t_m)) * 1.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-161], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.20000000000000036e-161

   1. Initial program 35.4%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

   8. Applied rewrites 83.8%

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

   if 7.20000000000000036e-161 < t

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 4: 93.9% accurate, 1.4× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 (tan k)) (* (/ (* (/ l k) (/ l k)) (* (sin k) t_m)) 1.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / tan(k)) * ((((l / k) * (l / k)) / (sin(k) * t_m)) * 1.0));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / Math.tan(k)) * ((((l / k) * (l / k)) / (Math.sin(k) * t_m)) * 1.0));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 / math.tan(k)) * ((((l / k) * (l / k)) / (math.sin(k) * t_m)) * 1.0))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 / tan(k)) * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(sin(k) * t_m)) * 1.0)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 / tan(k)) * ((((l / k) * (l / k)) / (sin(k) * t_m)) * 1.0));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Applied rewrites 29.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 38.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. pow-div N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

7. Applied rewrites 84.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 93.9

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

9. Applied rewrites 93.9%

   \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{\sin k \cdot t} \cdot 1\right) \]
10. Add Preprocessing

Alternative 5: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t_3 := \frac{2}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \left(\frac{\ell}{t\_2} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \frac{\frac{\ell}{k} \cdot \ell}{k \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m)) (t_3 (/ 2.0 (tan k))))
   (*
    t_s
    (if (<= k 1.8e+134)
      (* (/ l (* k k)) (* (/ l t_2) t_3))
      (* t_3 (/ (* (/ l k) l) (* k t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * t_m;
	double t_3 = 2.0 / tan(k);
	double tmp;
	if (k <= 1.8e+134) {
		tmp = (l / (k * k)) * ((l / t_2) * t_3);
	} else {
		tmp = t_3 * (((l / k) * l) / (k * t_2));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sin(k) * t_m
    t_3 = 2.0d0 / tan(k)
    if (k <= 1.8d+134) then
        tmp = (l / (k * k)) * ((l / t_2) * t_3)
    else
        tmp = t_3 * (((l / k) * l) / (k * t_2))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * t_m;
	double t_3 = 2.0 / Math.tan(k);
	double tmp;
	if (k <= 1.8e+134) {
		tmp = (l / (k * k)) * ((l / t_2) * t_3);
	} else {
		tmp = t_3 * (((l / k) * l) / (k * t_2));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sin(k) * t_m
	t_3 = 2.0 / math.tan(k)
	tmp = 0
	if k <= 1.8e+134:
		tmp = (l / (k * k)) * ((l / t_2) * t_3)
	else:
		tmp = t_3 * (((l / k) * l) / (k * t_2))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	t_3 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (k <= 1.8e+134)
		tmp = Float64(Float64(l / Float64(k * k)) * Float64(Float64(l / t_2) * t_3));
	else
		tmp = Float64(t_3 * Float64(Float64(Float64(l / k) * l) / Float64(k * t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) * t_m;
	t_3 = 2.0 / tan(k);
	tmp = 0.0;
	if (k <= 1.8e+134)
		tmp = (l / (k * k)) * ((l / t_2) * t_3);
	else
		tmp = t_3 * (((l / k) * l) / (k * t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.8e+134], N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.79999999999999994e134

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.1

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

   9. Applied rewrites 87.1%

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

   if 1.79999999999999994e134 < k

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 84.4

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 88.8

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

   9. Applied rewrites 88.8%

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

Alternative 6: 90.3% accurate, 1.4× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 (tan k)) (/ (* (/ l k) (/ l (* (sin k) t_m))) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / tan(k)) * (((l / k) * (l / (sin(k) * t_m))) / k));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 / tan(k)) * (((l / k) * (l / (sin(k) * t_m))) / k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / Math.tan(k)) * (((l / k) * (l / (Math.sin(k) * t_m))) / k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 / math.tan(k)) * (((l / k) * (l / (math.sin(k) * t_m))) / k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 / tan(k)) * Float64(Float64(Float64(l / k) * Float64(l / Float64(sin(k) * t_m))) / k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 / tan(k)) * (((l / k) * (l / (sin(k) * t_m))) / k));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Applied rewrites 29.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 38.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. pow-div N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

7. Applied rewrites 84.4%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 84.4

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 93.6

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

9. Applied rewrites 93.6%

   \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{\sin k \cdot t}}{k}} \]
10. Add Preprocessing

Alternative 7: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{k \cdot k} \cdot \ell}{\sin k \cdot t\_m} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\tan k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \sin k}\right) \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

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 5.8e-73)
    (* (/ 2.0 k) (* (/ (* (/ l (* k k)) l) (* (sin k) t_m)) 1.0))
    (* (* (/ 2.0 (tan k)) (/ l (* (* k k) (sin k)))) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-73) {
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (sin(k) * t_m)) * 1.0);
	} else {
		tmp = ((2.0 / tan(k)) * (l / ((k * k) * sin(k)))) * (l / 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 <= 5.8d-73) then
        tmp = (2.0d0 / k) * ((((l / (k * k)) * l) / (sin(k) * t_m)) * 1.0d0)
    else
        tmp = ((2.0d0 / tan(k)) * (l / ((k * k) * sin(k)))) * (l / 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 <= 5.8e-73) {
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (Math.sin(k) * t_m)) * 1.0);
	} else {
		tmp = ((2.0 / Math.tan(k)) * (l / ((k * k) * Math.sin(k)))) * (l / 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 <= 5.8e-73:
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (math.sin(k) * t_m)) * 1.0)
	else:
		tmp = ((2.0 / math.tan(k)) * (l / ((k * k) * math.sin(k)))) * (l / 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 <= 5.8e-73)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(Float64(l / Float64(k * k)) * l) / Float64(sin(k) * t_m)) * 1.0));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) * Float64(l / Float64(Float64(k * k) * sin(k)))) * Float64(l / 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 <= 5.8e-73)
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (sin(k) * t_m)) * 1.0);
	else
		tmp = ((2.0 / tan(k)) * (l / ((k * k) * sin(k)))) * (l / 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, 5.8e-73], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e-73

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 73.9%

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

   if 5.8e-73 < k

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 82.7

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

   9. Applied rewrites 82.7%

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

Alternative 8: 84.0% accurate, 1.4× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* k k)) (* (/ l (* (sin k) t_m)) (/ 2.0 (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (k * k)) * ((l / (sin(k) * t_m)) * (2.0 / tan(k))));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (k * k)) * ((l / (sin(k) * t_m)) * (2.0d0 / tan(k))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (k * k)) * ((l / (Math.sin(k) * t_m)) * (2.0 / Math.tan(k))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (k * k)) * ((l / (math.sin(k) * t_m)) * (2.0 / math.tan(k))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(k * k)) * Float64(Float64(l / Float64(sin(k) * t_m)) * Float64(2.0 / tan(k)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (k * k)) * ((l / (sin(k) * t_m)) * (2.0 / tan(k))));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Applied rewrites 29.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 38.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. pow-div N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

7. Applied rewrites 84.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 87.1

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

9. Applied rewrites 87.1%

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

Alternative 9: 80.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{k \cdot k} \cdot \ell}{\sin k \cdot t\_m} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \sin k}\right) \cdot 2}{t\_m \cdot \tan k}\\ \end{array} \end{array} \]

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 7.2e-76)
    (* (/ 2.0 k) (* (/ (* (/ l (* k k)) l) (* (sin k) t_m)) 1.0))
    (/ (* (* l (/ l (* (* k k) (sin k)))) 2.0) (* t_m (tan 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 <= 7.2e-76) {
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (sin(k) * t_m)) * 1.0);
	} else {
		tmp = ((l * (l / ((k * k) * sin(k)))) * 2.0) / (t_m * tan(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 <= 7.2d-76) then
        tmp = (2.0d0 / k) * ((((l / (k * k)) * l) / (sin(k) * t_m)) * 1.0d0)
    else
        tmp = ((l * (l / ((k * k) * sin(k)))) * 2.0d0) / (t_m * tan(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 <= 7.2e-76) {
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (Math.sin(k) * t_m)) * 1.0);
	} else {
		tmp = ((l * (l / ((k * k) * Math.sin(k)))) * 2.0) / (t_m * Math.tan(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 <= 7.2e-76:
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (math.sin(k) * t_m)) * 1.0)
	else:
		tmp = ((l * (l / ((k * k) * math.sin(k)))) * 2.0) / (t_m * math.tan(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 <= 7.2e-76)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(Float64(l / Float64(k * k)) * l) / Float64(sin(k) * t_m)) * 1.0));
	else
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(Float64(k * k) * sin(k)))) * 2.0) / Float64(t_m * tan(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 <= 7.2e-76)
		tmp = (2.0 / k) * ((((l / (k * k)) * l) / (sin(k) * t_m)) * 1.0);
	else
		tmp = ((l * (l / ((k * k) * sin(k)))) * 2.0) / (t_m * tan(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, 7.2e-76], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / N[(N[(k * k), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.2000000000000001e-76

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 73.9%

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

   if 7.2000000000000001e-76 < k

   1. Initial program 35.4%

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

   3. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 38.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow-div N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

   7. Applied rewrites 84.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   9. Applied rewrites 82.6%

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

Alternative 10: 79.2% accurate, 1.4× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (* (/ l (* k k)) l) 2.0) (* (* (sin k) t_m) (tan k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((((l / (k * k)) * l) * 2.0) / ((sin(k) * t_m) * tan(k)));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((((l / (k * k)) * l) * 2.0d0) / ((sin(k) * t_m) * tan(k)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((((l / (k * k)) * l) * 2.0) / ((Math.sin(k) * t_m) * Math.tan(k)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((((l / (k * k)) * l) * 2.0) / ((math.sin(k) * t_m) * math.tan(k)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(Float64(l / Float64(k * k)) * l) * 2.0) / Float64(Float64(sin(k) * t_m) * tan(k))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((((l / (k * k)) * l) * 2.0) / ((sin(k) * t_m) * tan(k)));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Applied rewrites 29.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 38.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. pow-div N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\sin k \cdot t} \cdot {t}^{\color{blue}{0}}\right) \]

7. Applied rewrites 84.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 84.0

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

9. Applied rewrites 84.0%

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

Alternative 11: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.06 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{2}{t\_m} \cdot \left(\frac{\ell}{k \cdot k} \cdot \ell\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\left(0.5 - 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\right)\right)\\ \end{array} \end{array} \]

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 (<= l 1.06e+158)
    (/ (* (/ 2.0 t_m) (* (/ l (* k k)) l)) (* k k))
    (* 2.0 (* l (* l (/ (cos k) (* (* (* (- 0.5 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 (l <= 1.06e+158) {
		tmp = ((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k);
	} else {
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - 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 (l <= 1.06d+158) then
        tmp = ((2.0d0 / t_m) * ((l / (k * k)) * l)) / (k * k)
    else
        tmp = 2.0d0 * (l * (l * (cos(k) / ((((0.5d0 - 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 (l <= 1.06e+158) {
		tmp = ((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k);
	} else {
		tmp = 2.0 * (l * (l * (Math.cos(k) / ((((0.5 - 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 l <= 1.06e+158:
		tmp = ((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k)
	else:
		tmp = 2.0 * (l * (l * (math.cos(k) / ((((0.5 - 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 (l <= 1.06e+158)
		tmp = Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(l / Float64(k * k)) * l)) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(Float64(Float64(0.5 - 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 (l <= 1.06e+158)
		tmp = ((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k);
	else
		tmp = 2.0 * (l * (l * (cos(k) / ((((0.5 - 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[l, 1.06e+158], N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.06e158

   1. Initial program 35.4%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 65.7

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

   9. Applied rewrites 65.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. times-frac N/A

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

       10. lift-pow.f64 N/A

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

       11. pow2 N/A

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

       12. associate-*l/ N/A

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

       13. lift-/.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 73.0

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

   11. Applied rewrites 73.0%

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

   if 1.06e158 < l

   1. Initial program 35.4%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 78.7%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 41.6%

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

Alternative 12: 73.0% accurate, 5.5× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (/ 2.0 t_m) (* (/ l (* k k)) l)) (* k k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((2.0d0 / t_m) * ((l / (k * k)) * l)) / (k * k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(2.0 / t_m) * Float64(Float64(l / Float64(k * k)) * l)) / Float64(k * k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((2.0 / t_m) * ((l / (k * k)) * l)) / (k * k));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 73.7

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

4. Applied rewrites 73.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 67.9%

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

7. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 65.7

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

9. Applied rewrites 65.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*r/ N/A

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

    4. lift-*.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. times-frac N/A

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

    10. lift-pow.f64 N/A

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

    11. pow2 N/A

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

    12. associate-*l/ N/A

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

    13. lift-/.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. lower-*.f64 N/A

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

    16. lower-/.f64 73.0

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

11. Applied rewrites 73.0%

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

Alternative 13: 72.9% accurate, 5.6× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (* l (/ l (* t_m (* k k)))) 2.0) (* k k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * (l / (t_m * (k * k)))) * 2.0) / (k * k));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l * (l / (t_m * (k * k)))) * 2.0d0) / (k * k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l * (l / (t_m * (k * k)))) * 2.0) / (k * k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l * (l / (t_m * (k * k)))) * 2.0) / (k * k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l * Float64(l / Float64(t_m * Float64(k * k)))) * 2.0) / Float64(k * k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l * (l / (t_m * (k * k)))) * 2.0) / (k * k));
end

Wolfram

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

Derivation

1. Initial program 35.4%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 73.7

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

4. Applied rewrites 73.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 67.9%

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

7. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 65.7

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

9. Applied rewrites 65.7%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 65.7

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

    4. lift-/.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. pow2 N/A

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

    7. associate-/l* N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 72.9

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

    10. lift-*.f64 N/A

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

    11. lift-pow.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f64 72.9

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

11. Applied rewrites 72.9%

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

Alternative 14: 65.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((((l + l) * l) / (t_m * (k * k))) / (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   7. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(\color{blue}{t} \cdot {\sin k}^{2}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)} \]

   9. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{\color{blue}{2}}\right)} \]

   10. lower-sin.f64 73.7

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

4. Applied rewrites 73.7%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}}{\color{blue}{{k}^{2}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {\sin k}^{2}}}{\color{blue}{{k}^{2}}} \]

6. Applied rewrites 67.9%

   \[\leadsto \frac{\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t}}{\color{blue}{k \cdot k}} \]

7. Taylor expanded in k around 0

   \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

   5. lower-pow.f64 65.7

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

9. Applied rewrites 65.7%

   \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\color{blue}{k} \cdot k} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

    5. lift-pow.f64 N/A

       \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}{k \cdot k} \]

    6. pow2 N/A

       \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t}}{k \cdot k} \]

    7. associate-*l* N/A

       \[\leadsto \frac{\frac{\left(2 \cdot \ell\right) \cdot \ell}{{k}^{2} \cdot t}}{k \cdot k} \]

    8. count-2 N/A

       \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{2} \cdot t}}{k \cdot k} \]

    9. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{2} \cdot t}}{k \cdot k} \]

    10. lower-*.f64 65.7

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{2} \cdot t}}{k \cdot k} \]

    11. lift-*.f64 N/A

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{2} \cdot t}}{k \cdot k} \]

    12. lift-pow.f64 N/A

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{{k}^{2} \cdot t}}{k \cdot k} \]

    13. pow2 N/A

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

    14. lift-*.f64 N/A

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot t}}{k \cdot k} \]

    15. *-commutative N/A

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k} \]

    16. lower-*.f64 65.7

        \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k} \]

11. Applied rewrites 65.7%

    \[\leadsto \frac{\frac{\left(\ell + \ell\right) \cdot \ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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))))