Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 92.4%

Time: 7.4s

Alternatives: 22

Speedup: 6.8×

• 73.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

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: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ t_2 := \frac{\ell}{\sin k \cdot \left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot \left|t\right|\right) \cdot \frac{k}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;\left|t\right| \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;\left(\frac{2}{\mathsf{fma}\left(\frac{k}{t\_1}, k, 2\right) \cdot \tan k} \cdot \frac{\ell}{t\_1}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot 2}{\left|t\right| \cdot \left(2 \cdot \tan k\right)} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t))) (t_2 (/ l (* (sin k) (fabs t)))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 4.8e-39)
     (/
      2.0
      (*
       (*
        (* (fma -0.5 (cos (+ k k)) 0.5) (fabs t))
        (/ k (* (cos k) l)))
       (/ k l)))
     (if (<= (fabs t) 1.35e+155)
       (* (* (/ 2.0 (* (fma (/ k t_1) k 2.0) (tan k))) (/ l t_1)) t_2)
       (*
        (/ (* (/ l (fabs t)) 2.0) (* (fabs t) (* 2.0 (tan k))))
        t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double t_2 = l / (sin(k) * fabs(t));
	double tmp;
	if (fabs(t) <= 4.8e-39) {
		tmp = 2.0 / (((fma(-0.5, cos((k + k)), 0.5) * fabs(t)) * (k / (cos(k) * l))) * (k / l));
	} else if (fabs(t) <= 1.35e+155) {
		tmp = ((2.0 / (fma((k / t_1), k, 2.0) * tan(k))) * (l / t_1)) * t_2;
	} else {
		tmp = (((l / fabs(t)) * 2.0) / (fabs(t) * (2.0 * tan(k)))) * t_2;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	t_2 = Float64(l / Float64(sin(k) * abs(t)))
	tmp = 0.0
	if (abs(t) <= 4.8e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(-0.5, cos(Float64(k + k)), 0.5) * abs(t)) * Float64(k / Float64(cos(k) * l))) * Float64(k / l)));
	elseif (abs(t) <= 1.35e+155)
		tmp = Float64(Float64(Float64(2.0 / Float64(fma(Float64(k / t_1), k, 2.0) * tan(k))) * Float64(l / t_1)) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(l / abs(t)) * 2.0) / Float64(abs(t) * Float64(2.0 * tan(k)))) * t_2);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4.8e-39], N[(2.0 / N[(N[(N[(N[(-0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.35e+155], N[(N[(N[(2.0 / N[(N[(N[(k / t$95$1), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.8000000000000003e-39

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 4.8000000000000003e-39 < t < 1.35e155

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

   if 1.35e155 < t

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.5%

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

Alternative 2: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot \left|t\right|\right) \cdot \frac{k}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;\left|t\right| \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot t\_1\right) \cdot \left(\frac{\left|t\right|}{\ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t\_1}, k, 2\right) \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot 2}{\left|t\right| \cdot \left(2 \cdot \tan k\right)} \cdot \frac{\ell}{\sin k \cdot \left|t\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 2.25e-80)
     (/
      2.0
      (*
       (*
        (* (fma -0.5 (cos (+ k k)) 0.5) (fabs t))
        (/ k (* (cos k) l)))
       (/ k l)))
     (if (<= (fabs t) 1.35e+155)
       (/
        2.0
        (*
         (* (/ (sin k) l) t_1)
         (* (/ (fabs t) l) (* (fma (/ k t_1) k 2.0) (tan k)))))
       (*
        (/ (* (/ l (fabs t)) 2.0) (* (fabs t) (* 2.0 (tan k))))
        (/ l (* (sin k) (fabs t)))))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double tmp;
	if (fabs(t) <= 2.25e-80) {
		tmp = 2.0 / (((fma(-0.5, cos((k + k)), 0.5) * fabs(t)) * (k / (cos(k) * l))) * (k / l));
	} else if (fabs(t) <= 1.35e+155) {
		tmp = 2.0 / (((sin(k) / l) * t_1) * ((fabs(t) / l) * (fma((k / t_1), k, 2.0) * tan(k))));
	} else {
		tmp = (((l / fabs(t)) * 2.0) / (fabs(t) * (2.0 * tan(k)))) * (l / (sin(k) * fabs(t)));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	tmp = 0.0
	if (abs(t) <= 2.25e-80)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(-0.5, cos(Float64(k + k)), 0.5) * abs(t)) * Float64(k / Float64(cos(k) * l))) * Float64(k / l)));
	elseif (abs(t) <= 1.35e+155)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * t_1) * Float64(Float64(abs(t) / l) * Float64(fma(Float64(k / t_1), k, 2.0) * tan(k)))));
	else
		tmp = Float64(Float64(Float64(Float64(l / abs(t)) * 2.0) / Float64(abs(t) * Float64(2.0 * tan(k)))) * Float64(l / Float64(sin(k) * abs(t))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.25e-80], N[(2.0 / N[(N[(N[(N[(-0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.35e+155], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k / t$95$1), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2500000000000001e-80

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 2.2500000000000001e-80 < t < 1.35e155

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 62.2%

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

   if 1.35e155 < t

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.5%

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

Alternative 3: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left|t\right| \cdot \left|t\right|\\ t_2 := \frac{\ell}{\sin k \cdot \left|t\right|}\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot \left|t\right|\right) \cdot \frac{k}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;\left|t\right| \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;\frac{\ell + \ell}{\left(\mathsf{fma}\left(\frac{k}{t\_1}, k, 2\right) \cdot \tan k\right) \cdot t\_1} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot 2}{\left|t\right| \cdot \left(2 \cdot \tan k\right)} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs t) (fabs t))) (t_2 (/ l (* (sin k) (fabs t)))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 4.8e-39)
     (/
      2.0
      (*
       (*
        (* (fma -0.5 (cos (+ k k)) 0.5) (fabs t))
        (/ k (* (cos k) l)))
       (/ k l)))
     (if (<= (fabs t) 1.35e+155)
       (* (/ (+ l l) (* (* (fma (/ k t_1) k 2.0) (tan k)) t_1)) t_2)
       (*
        (/ (* (/ l (fabs t)) 2.0) (* (fabs t) (* 2.0 (tan k))))
        t_2))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(t) * fabs(t);
	double t_2 = l / (sin(k) * fabs(t));
	double tmp;
	if (fabs(t) <= 4.8e-39) {
		tmp = 2.0 / (((fma(-0.5, cos((k + k)), 0.5) * fabs(t)) * (k / (cos(k) * l))) * (k / l));
	} else if (fabs(t) <= 1.35e+155) {
		tmp = ((l + l) / ((fma((k / t_1), k, 2.0) * tan(k)) * t_1)) * t_2;
	} else {
		tmp = (((l / fabs(t)) * 2.0) / (fabs(t) * (2.0 * tan(k)))) * t_2;
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(abs(t) * abs(t))
	t_2 = Float64(l / Float64(sin(k) * abs(t)))
	tmp = 0.0
	if (abs(t) <= 4.8e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(-0.5, cos(Float64(k + k)), 0.5) * abs(t)) * Float64(k / Float64(cos(k) * l))) * Float64(k / l)));
	elseif (abs(t) <= 1.35e+155)
		tmp = Float64(Float64(Float64(l + l) / Float64(Float64(fma(Float64(k / t_1), k, 2.0) * tan(k)) * t_1)) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(l / abs(t)) * 2.0) / Float64(abs(t) * Float64(2.0 * tan(k)))) * t_2);
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4.8e-39], N[(2.0 / N[(N[(N[(N[(-0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.35e+155], N[(N[(N[(l + l), $MachinePrecision] / N[(N[(N[(N[(k / t$95$1), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.8000000000000003e-39

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 4.8000000000000003e-39 < t < 1.35e155

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 61.4%

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

   7. Applied rewrites 61.4%

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

   if 1.35e155 < t

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.5%

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

Alternative 4: 91.8% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(-0.5, \cos \left(k + k\right), 0.5\right) \cdot \left|t\right|\right) \cdot \frac{k}{\cos k \cdot \ell}\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot 2}{\left|t\right| \cdot \left(\mathsf{fma}\left(\frac{k}{\left|t\right| \cdot \left|t\right|}, k, 2\right) \cdot \tan k\right)} \cdot \frac{\ell}{\sin k \cdot \left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 4.8e-39)
   (/
    2.0
    (*
     (* (* (fma -0.5 (cos (+ k k)) 0.5) (fabs t)) (/ k (* (cos k) l)))
     (/ k l)))
   (*
    (/
     (* (/ l (fabs t)) 2.0)
     (* (fabs t) (* (fma (/ k (* (fabs t) (fabs t))) k 2.0) (tan k))))
    (/ l (* (sin k) (fabs t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 4.8e-39) {
		tmp = 2.0 / (((fma(-0.5, cos((k + k)), 0.5) * fabs(t)) * (k / (cos(k) * l))) * (k / l));
	} else {
		tmp = (((l / fabs(t)) * 2.0) / (fabs(t) * (fma((k / (fabs(t) * fabs(t))), k, 2.0) * tan(k)))) * (l / (sin(k) * fabs(t)));
	}
	return copysign(1.0, t) * tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 4.8e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(-0.5, cos(Float64(k + k)), 0.5) * abs(t)) * Float64(k / Float64(cos(k) * l))) * Float64(k / l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / abs(t)) * 2.0) / Float64(abs(t) * Float64(fma(Float64(k / Float64(abs(t) * abs(t))), k, 2.0) * tan(k)))) * Float64(l / Float64(sin(k) * abs(t))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 4.8e-39], N[(2.0 / N[(N[(N[(N[(-0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(k / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.8000000000000003e-39

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

   if 4.8000000000000003e-39 < t

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 5: 90.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{t \cdot \left(2 \cdot \tan \left(\left|k\right|\right)\right)} \cdot \frac{\ell}{\sin \left(\left|k\right|\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(-0.5, \cos \left(\left|k\right| + \left|k\right|\right), 0.5\right) \cdot t\right) \cdot \frac{\left|k\right|}{\cos \left(\left|k\right|\right) \cdot \ell}\right) \cdot \frac{\left|k\right|}{\ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 6e+35)
  (*
   (/ (* (/ l t) 2.0) (* t (* 2.0 (tan (fabs k)))))
   (/ l (* (sin (fabs k)) t)))
  (/
   2.0
   (*
    (*
     (* (fma -0.5 (cos (+ (fabs k) (fabs k))) 0.5) t)
     (/ (fabs k) (* (cos (fabs k)) l)))
    (/ (fabs k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 6e+35) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * tan(fabs(k))))) * (l / (sin(fabs(k)) * t));
	} else {
		tmp = 2.0 / (((fma(-0.5, cos((fabs(k) + fabs(k))), 0.5) * t) * (fabs(k) / (cos(fabs(k)) * l))) * (fabs(k) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 6e+35)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) / Float64(t * Float64(2.0 * tan(abs(k))))) * Float64(l / Float64(sin(abs(k)) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(-0.5, cos(Float64(abs(k) + abs(k))), 0.5) * t) * Float64(abs(k) / Float64(cos(abs(k)) * l))) * Float64(abs(k) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 6e+35], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t * N[(2.0 * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(-0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.9999999999999998e35

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.5%

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

   if 5.9999999999999998e35 < k

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

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

Alternative 6: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|k\right| \leq 5.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{t \cdot \left(2 \cdot \tan \left(\left|k\right|\right)\right)} \cdot \frac{\ell}{\sin \left(\left|k\right|\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(-0.5, \cos \left(\left|k\right| + \left|k\right|\right), 0.5\right) \cdot \left(\left(\left|k\right| \cdot t\right) \cdot \frac{\left|k\right|}{\left(\cos \left(\left|k\right|\right) \cdot \ell\right) \cdot \ell}\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (if (<= (fabs k) 5.8e+35)
  (*
   (/ (* (/ l t) 2.0) (* t (* 2.0 (tan (fabs k)))))
   (/ l (* (sin (fabs k)) t)))
  (/
   2.0
   (*
    (fma -0.5 (cos (+ (fabs k) (fabs k))) 0.5)
    (* (* (fabs k) t) (/ (fabs k) (* (* (cos (fabs k)) l) l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(k) <= 5.8e+35) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * tan(fabs(k))))) * (l / (sin(fabs(k)) * t));
	} else {
		tmp = 2.0 / (fma(-0.5, cos((fabs(k) + fabs(k))), 0.5) * ((fabs(k) * t) * (fabs(k) / ((cos(fabs(k)) * l) * l))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(k) <= 5.8e+35)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) / Float64(t * Float64(2.0 * tan(abs(k))))) * Float64(l / Float64(sin(abs(k)) * t)));
	else
		tmp = Float64(2.0 / Float64(fma(-0.5, cos(Float64(abs(k) + abs(k))), 0.5) * Float64(Float64(abs(k) * t) * Float64(abs(k) / Float64(Float64(cos(abs(k)) * l) * l)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[Abs[k], $MachinePrecision], 5.8e+35], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t * N[(2.0 * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(-0.5 * N[Cos[N[(N[Abs[k], $MachinePrecision] + N[Abs[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] / N[(N[(N[Cos[N[Abs[k], $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.7999999999999999e35

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.5%

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

   if 5.7999999999999999e35 < k

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 60.2%

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

   8. Applied rewrites 60.2%

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

Alternative 7: 83.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \tan \left(\left|k\right|\right)\\ t_2 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{t \cdot \left(2 \cdot t\_1\right)} \cdot \frac{\ell}{t\_2 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left|k\right| \cdot t\right) \cdot \left|k\right|}{\ell \cdot \ell} \cdot \left(t\_2 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (tan (fabs k))) (t_2 (sin (fabs k))))
  (if (<= (fabs k) 6e+35)
    (* (/ (* (/ l t) 2.0) (* t (* 2.0 t_1))) (/ l (* t_2 t)))
    (/ 2.0 (* (/ (* (* (fabs k) t) (fabs k)) (* l l)) (* t_2 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(fabs(k));
	double t_2 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 6e+35) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * t_1))) * (l / (t_2 * t));
	} else {
		tmp = 2.0 / ((((fabs(k) * t) * fabs(k)) / (l * l)) * (t_2 * t_1));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = tan(abs(k))
    t_2 = sin(abs(k))
    if (abs(k) <= 6d+35) then
        tmp = (((l / t) * 2.0d0) / (t * (2.0d0 * t_1))) * (l / (t_2 * t))
    else
        tmp = 2.0d0 / ((((abs(k) * t) * abs(k)) / (l * l)) * (t_2 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(Math.abs(k));
	double t_2 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 6e+35) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * t_1))) * (l / (t_2 * t));
	} else {
		tmp = 2.0 / ((((Math.abs(k) * t) * Math.abs(k)) / (l * l)) * (t_2 * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(math.fabs(k))
	t_2 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 6e+35:
		tmp = (((l / t) * 2.0) / (t * (2.0 * t_1))) * (l / (t_2 * t))
	else:
		tmp = 2.0 / ((((math.fabs(k) * t) * math.fabs(k)) / (l * l)) * (t_2 * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = tan(abs(k))
	t_2 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 6e+35)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) / Float64(t * Float64(2.0 * t_1))) * Float64(l / Float64(t_2 * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) * abs(k)) / Float64(l * l)) * Float64(t_2 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(abs(k));
	t_2 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 6e+35)
		tmp = (((l / t) * 2.0) / (t * (2.0 * t_1))) * (l / (t_2 * t));
	else
		tmp = 2.0 / ((((abs(k) * t) * abs(k)) / (l * l)) * (t_2 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 6e+35], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.9999999999999998e35

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 70.5%

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

   if 5.9999999999999998e35 < k

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 64.3%

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

Alternative 8: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{t \cdot \left(2 \cdot \left|k\right|\right)} \cdot \frac{\ell}{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left|k\right| \cdot t\right) \cdot \left|k\right|}{\ell \cdot \ell} \cdot \left(t\_1 \cdot \tan \left(\left|k\right|\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (sin (fabs k))))
  (if (<= (fabs k) 1.02e-5)
    (* (/ (* (/ l t) 2.0) (* t (* 2.0 (fabs k)))) (/ l (* t_1 t)))
    (/
     2.0
     (*
      (/ (* (* (fabs k) t) (fabs k)) (* l l))
      (* t_1 (tan (fabs k))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 1.02e-5) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * fabs(k)))) * (l / (t_1 * t));
	} else {
		tmp = 2.0 / ((((fabs(k) * t) * fabs(k)) / (l * l)) * (t_1 * tan(fabs(k))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k))
    if (abs(k) <= 1.02d-5) then
        tmp = (((l / t) * 2.0d0) / (t * (2.0d0 * abs(k)))) * (l / (t_1 * t))
    else
        tmp = 2.0d0 / ((((abs(k) * t) * abs(k)) / (l * l)) * (t_1 * tan(abs(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 1.02e-5) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * Math.abs(k)))) * (l / (t_1 * t));
	} else {
		tmp = 2.0 / ((((Math.abs(k) * t) * Math.abs(k)) / (l * l)) * (t_1 * Math.tan(Math.abs(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 1.02e-5:
		tmp = (((l / t) * 2.0) / (t * (2.0 * math.fabs(k)))) * (l / (t_1 * t))
	else:
		tmp = 2.0 / ((((math.fabs(k) * t) * math.fabs(k)) / (l * l)) * (t_1 * math.tan(math.fabs(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 1.02e-5)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) / Float64(t * Float64(2.0 * abs(k)))) * Float64(l / Float64(t_1 * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(abs(k) * t) * abs(k)) / Float64(l * l)) * Float64(t_1 * tan(abs(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 1.02e-5)
		tmp = (((l / t) * 2.0) / (t * (2.0 * abs(k)))) * (l / (t_1 * t));
	else
		tmp = 2.0 / ((((abs(k) * t) * abs(k)) / (l * l)) * (t_1 * tan(abs(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.02e-5], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t * N[(2.0 * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0200000000000001e-5

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 68.5%

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

   10. Applied rewrites 68.5%

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

   if 1.0200000000000001e-5 < k

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 64.3%

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

Alternative 9: 82.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|k\right|\right)\\ \mathbf{if}\;\left|k\right| \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{t \cdot \left(2 \cdot \left|k\right|\right)} \cdot \frac{\ell}{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left|k\right| \cdot \left(\left|k\right| \cdot \left(\left(\tan \left(\left|k\right|\right) \cdot t\_1\right) \cdot \frac{t}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (sin (fabs k))))
  (if (<= (fabs k) 1.02e-5)
    (* (/ (* (/ l t) 2.0) (* t (* 2.0 (fabs k)))) (/ l (* t_1 t)))
    (/
     2.0
     (*
      (fabs k)
      (* (fabs k) (* (* (tan (fabs k)) t_1) (/ t (* l l)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(fabs(k));
	double tmp;
	if (fabs(k) <= 1.02e-5) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * fabs(k)))) * (l / (t_1 * t));
	} else {
		tmp = 2.0 / (fabs(k) * (fabs(k) * ((tan(fabs(k)) * t_1) * (t / (l * l)))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(abs(k))
    if (abs(k) <= 1.02d-5) then
        tmp = (((l / t) * 2.0d0) / (t * (2.0d0 * abs(k)))) * (l / (t_1 * t))
    else
        tmp = 2.0d0 / (abs(k) * (abs(k) * ((tan(abs(k)) * t_1) * (t / (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(Math.abs(k));
	double tmp;
	if (Math.abs(k) <= 1.02e-5) {
		tmp = (((l / t) * 2.0) / (t * (2.0 * Math.abs(k)))) * (l / (t_1 * t));
	} else {
		tmp = 2.0 / (Math.abs(k) * (Math.abs(k) * ((Math.tan(Math.abs(k)) * t_1) * (t / (l * l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(math.fabs(k))
	tmp = 0
	if math.fabs(k) <= 1.02e-5:
		tmp = (((l / t) * 2.0) / (t * (2.0 * math.fabs(k)))) * (l / (t_1 * t))
	else:
		tmp = 2.0 / (math.fabs(k) * (math.fabs(k) * ((math.tan(math.fabs(k)) * t_1) * (t / (l * l)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(abs(k))
	tmp = 0.0
	if (abs(k) <= 1.02e-5)
		tmp = Float64(Float64(Float64(Float64(l / t) * 2.0) / Float64(t * Float64(2.0 * abs(k)))) * Float64(l / Float64(t_1 * t)));
	else
		tmp = Float64(2.0 / Float64(abs(k) * Float64(abs(k) * Float64(Float64(tan(abs(k)) * t_1) * Float64(t / Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(abs(k));
	tmp = 0.0;
	if (abs(k) <= 1.02e-5)
		tmp = (((l / t) * 2.0) / (t * (2.0 * abs(k)))) * (l / (t_1 * t));
	else
		tmp = 2.0 / (abs(k) * (abs(k) * ((tan(abs(k)) * t_1) * (t / (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sin[N[Abs[k], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 1.02e-5], N[(N[(N[(N[(l / t), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t * N[(2.0 * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Abs[k], $MachinePrecision] * N[(N[Abs[k], $MachinePrecision] * N[(N[(N[Tan[N[Abs[k], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0200000000000001e-5

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 68.5%

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

   10. Applied rewrites 68.5%

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

   if 1.0200000000000001e-5 < k

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

   6. Applied rewrites 64.7%

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

Alternative 10: 75.7% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 5.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{\left|t\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left|t\right|} \cdot 2}{\left|t\right| \cdot \left(2 \cdot k\right)} \cdot \frac{\ell}{\sin k \cdot \left|t\right|}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (*
 (copysign 1.0 t)
 (if (<= (fabs t) 5.8e-32)
   (/ 2.0 (* (/ (pow k 4.0) l) (/ (fabs t) l)))
   (*
    (/ (* (/ l (fabs t)) 2.0) (* (fabs t) (* 2.0 k)))
    (/ l (* (sin k) (fabs t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (fabs(t) <= 5.8e-32) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (fabs(t) / l));
	} else {
		tmp = (((l / fabs(t)) * 2.0) / (fabs(t) * (2.0 * k))) * (l / (sin(k) * fabs(t)));
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (Math.abs(t) <= 5.8e-32) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (Math.abs(t) / l));
	} else {
		tmp = (((l / Math.abs(t)) * 2.0) / (Math.abs(t) * (2.0 * k))) * (l / (Math.sin(k) * Math.abs(t)));
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if math.fabs(t) <= 5.8e-32:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (math.fabs(t) / l))
	else:
		tmp = (((l / math.fabs(t)) * 2.0) / (math.fabs(t) * (2.0 * k))) * (l / (math.sin(k) * math.fabs(t)))
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (abs(t) <= 5.8e-32)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(abs(t) / l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / abs(t)) * 2.0) / Float64(abs(t) * Float64(2.0 * k))) * Float64(l / Float64(sin(k) * abs(t))));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (abs(t) <= 5.8e-32)
		tmp = 2.0 / (((k ^ 4.0) / l) * (abs(t) / l));
	else
		tmp = (((l / abs(t)) * 2.0) / (abs(t) * (2.0 * k))) * (l / (sin(k) * abs(t)));
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 5.8e-32], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999999e-32

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 51.9%

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

   7. Applied rewrites 51.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 56.2%

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

   9. Applied rewrites 56.2%

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

   if 5.7999999999999999e-32 < t

   1. Initial program 54.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 66.5%

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

   3. Applied rewrites 66.5%

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

   5. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 73.0%

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 68.5%

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

   10. Applied rewrites 68.5%

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

Alternative 11: 74.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.1 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{\left|t\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left|t\right|}}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 2.1e-38)
     (/ 2.0 (* (/ (pow k 4.0) l) (/ (fabs t) l)))
     (* l (/ (/ (/ l (fabs t)) t_1) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 2.1e-38) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (fabs(t) / l));
	} else {
		tmp = l * (((l / fabs(t)) / t_1) / t_1);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 2.1e-38) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (Math.abs(t) / l));
	} else {
		tmp = l * (((l / Math.abs(t)) / t_1) / t_1);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 2.1e-38:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (math.fabs(t) / l))
	else:
		tmp = l * (((l / math.fabs(t)) / t_1) / t_1)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 2.1e-38)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(abs(t) / l)));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / abs(t)) / t_1) / t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 2.1e-38)
		tmp = 2.0 / (((k ^ 4.0) / l) * (abs(t) / l));
	else
		tmp = l * (((l / abs(t)) / t_1) / t_1);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.1e-38], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1000000000000001e-38

   1. Initial program 54.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 60.5%

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

   4. Applied rewrites 60.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 51.9%

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

   7. Applied rewrites 51.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 56.2%

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

   9. Applied rewrites 56.2%

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

   if 2.1000000000000001e-38 < t

   1. Initial program 54.7%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 51.2%

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

   4. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.1%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 59.5%

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

      14. lift-pow.f64 N/A

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

      15. unpow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 59.5%

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

   6. Applied rewrites 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ t_2 := -\left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.75 \cdot 10^{-205}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left|t\right| \cdot \left(\left(\left(k \cdot k\right) \cdot t\_2\right) \cdot t\_2\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{\left|t\right|}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left|t\right|}}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t))) (t_2 (- (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.75e-205)
     (* l (/ l (* (fabs t) (* (* (* k k) t_2) t_2))))
     (if (<= (fabs t) 2e-38)
       (/ 2.0 (* (pow k 4.0) (/ (fabs t) (* l l))))
       (* l (/ (/ (/ l (fabs t)) t_1) t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double t_2 = -fabs(t);
	double tmp;
	if (fabs(t) <= 1.75e-205) {
		tmp = l * (l / (fabs(t) * (((k * k) * t_2) * t_2)));
	} else if (fabs(t) <= 2e-38) {
		tmp = 2.0 / (pow(k, 4.0) * (fabs(t) / (l * l)));
	} else {
		tmp = l * (((l / fabs(t)) / t_1) / t_1);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double t_2 = -Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.75e-205) {
		tmp = l * (l / (Math.abs(t) * (((k * k) * t_2) * t_2)));
	} else if (Math.abs(t) <= 2e-38) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (Math.abs(t) / (l * l)));
	} else {
		tmp = l * (((l / Math.abs(t)) / t_1) / t_1);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	t_2 = -math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.75e-205:
		tmp = l * (l / (math.fabs(t) * (((k * k) * t_2) * t_2)))
	elif math.fabs(t) <= 2e-38:
		tmp = 2.0 / (math.pow(k, 4.0) * (math.fabs(t) / (l * l)))
	else:
		tmp = l * (((l / math.fabs(t)) / t_1) / t_1)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	t_2 = Float64(-abs(t))
	tmp = 0.0
	if (abs(t) <= 1.75e-205)
		tmp = Float64(l * Float64(l / Float64(abs(t) * Float64(Float64(Float64(k * k) * t_2) * t_2))));
	elseif (abs(t) <= 2e-38)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(abs(t) / Float64(l * l))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / abs(t)) / t_1) / t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	t_2 = -abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.75e-205)
		tmp = l * (l / (abs(t) * (((k * k) * t_2) * t_2)));
	elseif (abs(t) <= 2e-38)
		tmp = 2.0 / ((k ^ 4.0) * (abs(t) / (l * l)));
	else
		tmp = l * (((l / abs(t)) / t_1) / t_1);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[t], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.75e-205], N[(l * N[(l / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2e-38], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(N[Abs[t], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.75e-205

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      4. sqr-neg-rev N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{t}\right)\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      9. lower-neg.f64 61.9%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \left(-t\right)\right)} \]

   10. Applied rewrites 61.9%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \color{blue}{\left(-t\right)}\right)} \]

   if 1.75e-205 < t < 1.9999999999999999e-38

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.5%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{\color{blue}{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      4. lower-pow.f64 51.9%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

   7. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{\color{blue}{2}}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}} \]

      8. lower-/.f64 52.8%

         \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\ell \cdot \color{blue}{\ell}}} \]

   9. Applied rewrites 52.8%

      \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}} \]

   if 1.9999999999999999e-38 < t

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 72.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ t_2 := -\left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.8 \cdot 10^{-213}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left|t\right| \cdot \left(\left(\left(k \cdot k\right) \cdot t\_2\right) \cdot t\_2\right)}\\ \mathbf{elif}\;\left|t\right| \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\left|t\right| \cdot \frac{{k}^{4}}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left|t\right|}}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t))) (t_2 (- (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 2.8e-213)
     (* l (/ l (* (fabs t) (* (* (* k k) t_2) t_2))))
     (if (<= (fabs t) 2e-38)
       (/ 2.0 (* (fabs t) (/ (pow k 4.0) (* l l))))
       (* l (/ (/ (/ l (fabs t)) t_1) t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double t_2 = -fabs(t);
	double tmp;
	if (fabs(t) <= 2.8e-213) {
		tmp = l * (l / (fabs(t) * (((k * k) * t_2) * t_2)));
	} else if (fabs(t) <= 2e-38) {
		tmp = 2.0 / (fabs(t) * (pow(k, 4.0) / (l * l)));
	} else {
		tmp = l * (((l / fabs(t)) / t_1) / t_1);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double t_2 = -Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 2.8e-213) {
		tmp = l * (l / (Math.abs(t) * (((k * k) * t_2) * t_2)));
	} else if (Math.abs(t) <= 2e-38) {
		tmp = 2.0 / (Math.abs(t) * (Math.pow(k, 4.0) / (l * l)));
	} else {
		tmp = l * (((l / Math.abs(t)) / t_1) / t_1);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	t_2 = -math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 2.8e-213:
		tmp = l * (l / (math.fabs(t) * (((k * k) * t_2) * t_2)))
	elif math.fabs(t) <= 2e-38:
		tmp = 2.0 / (math.fabs(t) * (math.pow(k, 4.0) / (l * l)))
	else:
		tmp = l * (((l / math.fabs(t)) / t_1) / t_1)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	t_2 = Float64(-abs(t))
	tmp = 0.0
	if (abs(t) <= 2.8e-213)
		tmp = Float64(l * Float64(l / Float64(abs(t) * Float64(Float64(Float64(k * k) * t_2) * t_2))));
	elseif (abs(t) <= 2e-38)
		tmp = Float64(2.0 / Float64(abs(t) * Float64((k ^ 4.0) / Float64(l * l))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / abs(t)) / t_1) / t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	t_2 = -abs(t);
	tmp = 0.0;
	if (abs(t) <= 2.8e-213)
		tmp = l * (l / (abs(t) * (((k * k) * t_2) * t_2)));
	elseif (abs(t) <= 2e-38)
		tmp = 2.0 / (abs(t) * ((k ^ 4.0) / (l * l)));
	else
		tmp = l * (((l / abs(t)) / t_1) / t_1);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[t], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 2.8e-213], N[(l * N[(l / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2e-38], N[(2.0 / N[(N[Abs[t], $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.8e-213

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      4. sqr-neg-rev N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{t}\right)\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      9. lower-neg.f64 61.9%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \left(-t\right)\right)} \]

   10. Applied rewrites 61.9%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \color{blue}{\left(-t\right)}\right)} \]

   if 2.8e-213 < t < 1.9999999999999999e-38

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.5%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{\color{blue}{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      4. lower-pow.f64 51.9%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

   7. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{\color{blue}{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\color{blue}{\ell \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\color{blue}{\ell \cdot \ell}}} \]

      9. lower-/.f64 52.2%

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\ell \cdot \color{blue}{\ell}}} \]

   9. Applied rewrites 52.2%

      \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\color{blue}{\ell \cdot \ell}}} \]

   if 1.9999999999999999e-38 < t

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\left|k\right|\right)}^{4} \cdot t}{\ell \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs k) t)))
  (if (<= (fabs k) 2.25e-5)
    (* l (/ (/ (/ l t) t_1) t_1))
    (/ 2.0 (/ (* (pow (fabs k) 4.0) t) (* l l))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 2.25e-5) {
		tmp = l * (((l / t) / t_1) / t_1);
	} else {
		tmp = 2.0 / ((pow(fabs(k), 4.0) * t) / (l * l));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 2.25d-5) then
        tmp = l * (((l / t) / t_1) / t_1)
    else
        tmp = 2.0d0 / (((abs(k) ** 4.0d0) * t) / (l * l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 2.25e-5) {
		tmp = l * (((l / t) / t_1) / t_1);
	} else {
		tmp = 2.0 / ((Math.pow(Math.abs(k), 4.0) * t) / (l * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 2.25e-5:
		tmp = l * (((l / t) / t_1) / t_1)
	else:
		tmp = 2.0 / ((math.pow(math.fabs(k), 4.0) * t) / (l * l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 2.25e-5)
		tmp = Float64(l * Float64(Float64(Float64(l / t) / t_1) / t_1));
	else
		tmp = Float64(2.0 / Float64(Float64((abs(k) ^ 4.0) * t) / Float64(l * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 2.25e-5)
		tmp = l * (((l / t) / t_1) / t_1);
	else
		tmp = 2.0 / (((abs(k) ^ 4.0) * t) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 2.25e-5], N[(l * N[(N[(N[(l / t), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Abs[k], $MachinePrecision], 4.0], $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2500000000000001e-5

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]

   if 2.2500000000000001e-5 < k

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2}} \cdot \cos k}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\color{blue}{\ell}}^{2} \cdot \cos k}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{\color{blue}{2}} \cdot \cos k}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \color{blue}{\cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos \color{blue}{k}}} \]

      9. lower-cos.f64 60.5%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 60.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{\color{blue}{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      4. lower-pow.f64 51.9%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

   7. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}} \]

      3. lift-*.f64 51.9%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}} \]

   9. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 71.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ t_2 := -\left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left|t\right| \cdot \left(\left(\left(k \cdot k\right) \cdot t\_2\right) \cdot t\_2\right)}\\ \mathbf{elif}\;\left|t\right| \leq 1.62 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right| \cdot \left|t\right|}}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left|t\right|}}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t))) (t_2 (- (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.55e-162)
     (* l (/ l (* (fabs t) (* (* (* k k) t_2) t_2))))
     (if (<= (fabs t) 1.62e-38)
       (/ (/ (* l (/ l (* k k))) (* (fabs t) (fabs t))) (fabs t))
       (* l (/ (/ (/ l (fabs t)) t_1) t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double t_2 = -fabs(t);
	double tmp;
	if (fabs(t) <= 1.55e-162) {
		tmp = l * (l / (fabs(t) * (((k * k) * t_2) * t_2)));
	} else if (fabs(t) <= 1.62e-38) {
		tmp = ((l * (l / (k * k))) / (fabs(t) * fabs(t))) / fabs(t);
	} else {
		tmp = l * (((l / fabs(t)) / t_1) / t_1);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double t_2 = -Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.55e-162) {
		tmp = l * (l / (Math.abs(t) * (((k * k) * t_2) * t_2)));
	} else if (Math.abs(t) <= 1.62e-38) {
		tmp = ((l * (l / (k * k))) / (Math.abs(t) * Math.abs(t))) / Math.abs(t);
	} else {
		tmp = l * (((l / Math.abs(t)) / t_1) / t_1);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	t_2 = -math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.55e-162:
		tmp = l * (l / (math.fabs(t) * (((k * k) * t_2) * t_2)))
	elif math.fabs(t) <= 1.62e-38:
		tmp = ((l * (l / (k * k))) / (math.fabs(t) * math.fabs(t))) / math.fabs(t)
	else:
		tmp = l * (((l / math.fabs(t)) / t_1) / t_1)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	t_2 = Float64(-abs(t))
	tmp = 0.0
	if (abs(t) <= 1.55e-162)
		tmp = Float64(l * Float64(l / Float64(abs(t) * Float64(Float64(Float64(k * k) * t_2) * t_2))));
	elseif (abs(t) <= 1.62e-38)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / Float64(abs(t) * abs(t))) / abs(t));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / abs(t)) / t_1) / t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	t_2 = -abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.55e-162)
		tmp = l * (l / (abs(t) * (((k * k) * t_2) * t_2)));
	elseif (abs(t) <= 1.62e-38)
		tmp = ((l * (l / (k * k))) / (abs(t) * abs(t))) / abs(t);
	else
		tmp = l * (((l / abs(t)) / t_1) / t_1);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[t], $MachinePrecision])}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.55e-162], N[(l * N[(l / N[(N[Abs[t], $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 1.62e-38], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.5499999999999999e-162

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      4. sqr-neg-rev N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t\right)\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{t}\right)\right)\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} \]

      9. lower-neg.f64 61.9%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \left(-t\right)\right)} \]

   10. Applied rewrites 61.9%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(-t\right)\right) \cdot \color{blue}{\left(-t\right)}\right)} \]

   if 1.5499999999999999e-162 < t < 1.62e-38

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. unpow3 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      16. lower-/.f64 57.9%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      18. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

      19. lift-*.f64 57.9%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

   6. Applied rewrites 57.9%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{\color{blue}{t}} \]

   if 1.62e-38 < t

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_1 := k \cdot \left|t\right|\\ \mathsf{copysign}\left(1, t\right) \cdot \begin{array}{l} \mathbf{if}\;\left|t\right| \leq 1.55 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{\left|t\right| \cdot \left|t\right|}}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left|t\right|}}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* k (fabs t))))
  (*
   (copysign 1.0 t)
   (if (<= (fabs t) 1.55e-46)
     (/ (/ (* l (/ l (* k k))) (* (fabs t) (fabs t))) (fabs t))
     (* l (/ (/ (/ l (fabs t)) t_1) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = k * fabs(t);
	double tmp;
	if (fabs(t) <= 1.55e-46) {
		tmp = ((l * (l / (k * k))) / (fabs(t) * fabs(t))) / fabs(t);
	} else {
		tmp = l * (((l / fabs(t)) / t_1) / t_1);
	}
	return copysign(1.0, t) * tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = k * Math.abs(t);
	double tmp;
	if (Math.abs(t) <= 1.55e-46) {
		tmp = ((l * (l / (k * k))) / (Math.abs(t) * Math.abs(t))) / Math.abs(t);
	} else {
		tmp = l * (((l / Math.abs(t)) / t_1) / t_1);
	}
	return Math.copySign(1.0, t) * tmp;
}

Python

def code(t, l, k):
	t_1 = k * math.fabs(t)
	tmp = 0
	if math.fabs(t) <= 1.55e-46:
		tmp = ((l * (l / (k * k))) / (math.fabs(t) * math.fabs(t))) / math.fabs(t)
	else:
		tmp = l * (((l / math.fabs(t)) / t_1) / t_1)
	return math.copysign(1.0, t) * tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * abs(t))
	tmp = 0.0
	if (abs(t) <= 1.55e-46)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) / Float64(abs(t) * abs(t))) / abs(t));
	else
		tmp = Float64(l * Float64(Float64(Float64(l / abs(t)) / t_1) / t_1));
	end
	return Float64(copysign(1.0, t) * tmp)
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * abs(t);
	tmp = 0.0;
	if (abs(t) <= 1.55e-46)
		tmp = ((l * (l / (k * k))) / (abs(t) * abs(t))) / abs(t);
	else
		tmp = l * (((l / abs(t)) / t_1) / t_1);
	end
	tmp_2 = (sign(t) * abs(1.0)) * tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[Abs[t], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[t], $MachinePrecision], 1.55e-46], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.55e-46

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{\color{blue}{3}}} \]

      8. unpow3 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot \color{blue}{t}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{\left(t \cdot t\right) \cdot t} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{\color{blue}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t \cdot t}}{t} \]

      14. associate-/l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      16. lower-/.f64 57.9%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{{k}^{2}}}{t \cdot t}}{t} \]

      18. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

      19. lift-*.f64 57.9%

          \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{t} \]

   6. Applied rewrites 57.9%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{t \cdot t}}{\color{blue}{t}} \]

   if 1.55e-46 < t

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 69.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 7.2 \cdot 10^{+35}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\left(t\_1 \cdot t\right) \cdot \left|k\right|}}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs k) t)))
  (if (<= (fabs k) 7.2e+35)
    (* l (/ (/ (/ l t) t_1) t_1))
    (/ (/ (* l l) (* (* t_1 t) (fabs k))) t))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 7.2e+35) {
		tmp = l * (((l / t) / t_1) / t_1);
	} else {
		tmp = ((l * l) / ((t_1 * t) * fabs(k))) / t;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 7.2d+35) then
        tmp = l * (((l / t) / t_1) / t_1)
    else
        tmp = ((l * l) / ((t_1 * t) * abs(k))) / t
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 7.2e+35) {
		tmp = l * (((l / t) / t_1) / t_1);
	} else {
		tmp = ((l * l) / ((t_1 * t) * Math.abs(k))) / t;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 7.2e+35:
		tmp = l * (((l / t) / t_1) / t_1)
	else:
		tmp = ((l * l) / ((t_1 * t) * math.fabs(k))) / t
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 7.2e+35)
		tmp = Float64(l * Float64(Float64(Float64(l / t) / t_1) / t_1));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(t_1 * t) * abs(k))) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 7.2e+35)
		tmp = l * (((l / t) / t_1) / t_1);
	else
		tmp = ((l * l) / ((t_1 * t) * abs(k))) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 7.2e+35], N[(l * N[(N[(N[(l / t), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$1 * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.2000000000000001e35

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      8. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      10. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t \cdot k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{t} \cdot k} \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      13. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{t \cdot k} \]

      14. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

      15. lift-*.f64 69.3%

          \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{k \cdot \color{blue}{t}} \]

   10. Applied rewrites 69.3%

       \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot t}}{\color{blue}{k \cdot t}} \]

   if 7.2000000000000001e35 < k

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t} \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{t}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{t}} \]

      9. lower-/.f64 54.5%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      13. unswap-sqr N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}}{t} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k}}{t} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k}}{t} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}}{t} \]

      18. lift-*.f64 62.7%

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}}{t} \]

   10. Applied rewrites 62.7%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 67.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\left|k\right| \cdot t\right) \cdot t\right) \cdot \left|k\right|\\ \mathbf{if}\;\left|k\right| \leq 7.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\ell}{t\_1} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_1}}{t}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (* (* (fabs k) t) t) (fabs k))))
  (if (<= (fabs k) 7.2e+35)
    (* (/ l t_1) (/ l t))
    (/ (/ (* l l) t_1) t))))

C

double code(double t, double l, double k) {
	double t_1 = ((fabs(k) * t) * t) * fabs(k);
	double tmp;
	if (fabs(k) <= 7.2e+35) {
		tmp = (l / t_1) * (l / t);
	} else {
		tmp = ((l * l) / t_1) / t;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((abs(k) * t) * t) * abs(k)
    if (abs(k) <= 7.2d+35) then
        tmp = (l / t_1) * (l / t)
    else
        tmp = ((l * l) / t_1) / t
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = ((Math.abs(k) * t) * t) * Math.abs(k);
	double tmp;
	if (Math.abs(k) <= 7.2e+35) {
		tmp = (l / t_1) * (l / t);
	} else {
		tmp = ((l * l) / t_1) / t;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = ((math.fabs(k) * t) * t) * math.fabs(k)
	tmp = 0
	if math.fabs(k) <= 7.2e+35:
		tmp = (l / t_1) * (l / t)
	else:
		tmp = ((l * l) / t_1) / t
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(Float64(abs(k) * t) * t) * abs(k))
	tmp = 0.0
	if (abs(k) <= 7.2e+35)
		tmp = Float64(Float64(l / t_1) * Float64(l / t));
	else
		tmp = Float64(Float64(Float64(l * l) / t_1) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = ((abs(k) * t) * t) * abs(k);
	tmp = 0.0;
	if (abs(k) <= 7.2e+35)
		tmp = (l / t_1) * (l / t);
	else
		tmp = ((l * l) / t_1) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 7.2e+35], N[(N[(l / t$95$1), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.2000000000000001e35

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      6. times-frac N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

      9. lower-/.f64 58.4%

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{\ell}}{t} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

      13. unswap-sqr N/A

          \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

      17. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

      18. lift-*.f64 66.7%

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

   10. Applied rewrites 66.7%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t}} \]

   if 7.2000000000000001e35 < k

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t} \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{t}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{\color{blue}{t}} \]

      9. lower-/.f64 54.5%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}}{t} \]

      13. unswap-sqr N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}}{t} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k}}{t} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k}}{t} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k}}{t} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}}{t} \]

      18. lift-*.f64 62.7%

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}}{t} \]

   10. Applied rewrites 62.7%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 67.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} t_1 := \left|k\right| \cdot t\\ \mathbf{if}\;\left|k\right| \leq 5 \cdot 10^{+169}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t}}{\left(t\_1 \cdot t\right) \cdot \left|k\right|}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t\_1 \cdot \left|k\right|\right) \cdot t\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (let* ((t_1 (* (fabs k) t)))
  (if (<= (fabs k) 5e+169)
    (* l (/ (/ l t) (* (* t_1 t) (fabs k))))
    (* l (/ l (* t (* (* t_1 (fabs k)) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = fabs(k) * t;
	double tmp;
	if (fabs(k) <= 5e+169) {
		tmp = l * ((l / t) / ((t_1 * t) * fabs(k)));
	} else {
		tmp = l * (l / (t * ((t_1 * fabs(k)) * t)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(k) * t
    if (abs(k) <= 5d+169) then
        tmp = l * ((l / t) / ((t_1 * t) * abs(k)))
    else
        tmp = l * (l / (t * ((t_1 * abs(k)) * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.abs(k) * t;
	double tmp;
	if (Math.abs(k) <= 5e+169) {
		tmp = l * ((l / t) / ((t_1 * t) * Math.abs(k)));
	} else {
		tmp = l * (l / (t * ((t_1 * Math.abs(k)) * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.fabs(k) * t
	tmp = 0
	if math.fabs(k) <= 5e+169:
		tmp = l * ((l / t) / ((t_1 * t) * math.fabs(k)))
	else:
		tmp = l * (l / (t * ((t_1 * math.fabs(k)) * t)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(abs(k) * t)
	tmp = 0.0
	if (abs(k) <= 5e+169)
		tmp = Float64(l * Float64(Float64(l / t) / Float64(Float64(t_1 * t) * abs(k))));
	else
		tmp = Float64(l * Float64(l / Float64(t * Float64(Float64(t_1 * abs(k)) * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = abs(k) * t;
	tmp = 0.0;
	if (abs(k) <= 5e+169)
		tmp = l * ((l / t) / ((t_1 * t) * abs(k)));
	else
		tmp = l * (l / (t * ((t_1 * abs(k)) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Abs[k], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[N[Abs[k], $MachinePrecision], 5e+169], N[(l * N[(N[(l / t), $MachinePrecision] / N[(N[(t$95$1 * t), $MachinePrecision] * N[Abs[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t * N[(N[(t$95$1 * N[Abs[k], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000002e169

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right)} \cdot \left(k \cdot k\right)} \]

      5. lower-/.f64 58.4%

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      9. unswap-sqr N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

      11. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \]

      13. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \]

      14. lift-*.f64 66.7%

          \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \]

   10. Applied rewrites 66.7%

       \[\leadsto \ell \cdot \frac{\frac{\ell}{t}}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k}} \]

   if 5.0000000000000002e169 < k

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      5. lower-pow.f64 51.2%

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      6. lower-/.f64 55.1%

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

      13. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

      15. unpow3 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      16. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      17. lower-*.f64 59.5%

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   6. Applied rewrites 59.5%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      9. lower-*.f64 58.1%

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   8. Applied rewrites 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{t}\right)} \]

      6. *-commutative N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right)} \]

      8. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(t \cdot k\right) \cdot k\right) \cdot t\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(t \cdot k\right) \cdot k\right) \cdot t\right)} \]

      10. *-commutative N/A

          \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right)} \]

      11. lift-*.f64 64.7%

          \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{t}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 66.7% accurate, 6.5× speedup

Math

\[\frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* (/ l (* (* (* k t) t) k)) (/ l t)))

C

double code(double t, double l, double k) {
	return (l / (((k * t) * t) * k)) * (l / t);
}

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 = (l / (((k * t) * t) * k)) * (l / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l / (((k * t) * t) * k)) * (l / t);
}

Python

def code(t, l, k):
	return (l / (((k * t) * t) * k)) * (l / t)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(Float64(Float64(k * t) * t) * k)) * Float64(l / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (((k * t) * t) * k)) * (l / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(N[(N[(k * t), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.2%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.1%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   5. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   7. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   9. lower-*.f64 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

8. Applied rewrites 58.1%

   \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{t}} \]

   6. times-frac N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

   9. lower-/.f64 58.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{\ell}}{t} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{t} \]

   13. unswap-sqr N/A

       \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t} \]

   14. associate-*r* N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

   17. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

   18. lift-*.f64 66.7%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

10. Applied rewrites 66.7%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t}} \]
11. Add Preprocessing

Alternative 21: 65.9% accurate, 6.8× speedup

Math

\[\ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right)} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* l (/ l (* t (* (* (* k t) k) t)))))

C

double code(double t, double l, double k) {
	return l * (l / (t * (((k * t) * k) * t)));
}

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 = l * (l / (t * (((k * t) * k) * t)))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l / (t * (((k * t) * k) * t)));
}

Python

def code(t, l, k):
	return l * (l / (t * (((k * t) * k) * t)))

Julia

function code(t, l, k)
	return Float64(l * Float64(l / Float64(t * Float64(Float64(Float64(k * t) * k) * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l / (t * (((k * t) * k) * t)));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l / N[(t * N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.2%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.1%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   5. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   7. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   9. lower-*.f64 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

8. Applied rewrites 58.1%

   \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   2. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{t}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{t}\right)} \]

   6. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right)} \]

   8. associate-*r* N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(t \cdot k\right) \cdot k\right) \cdot t\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(t \cdot k\right) \cdot k\right) \cdot t\right)} \]

   10. *-commutative N/A

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right)} \]

   11. lift-*.f64 64.7%

       \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right)} \]

10. Applied rewrites 64.7%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{t}\right)} \]
11. Add Preprocessing

Alternative 22: 64.7% accurate, 6.8× speedup

Math

\[\ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \]

FPCore

(FPCore (t l k)
  :precision binary64
  (* l (/ l (* t (* (* k t) (* k t))))))

C

double code(double t, double l, double k) {
	return l * (l / (t * ((k * t) * (k * t))));
}

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 = l * (l / (t * ((k * t) * (k * t))))
end function

Java

public static double code(double t, double l, double k) {
	return l * (l / (t * ((k * t) * (k * t))));
}

Python

def code(t, l, k):
	return l * (l / (t * ((k * t) * (k * t))))

Julia

function code(t, l, k)
	return Float64(l * Float64(l / Float64(t * Float64(Float64(k * t) * Float64(k * t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (l / (t * ((k * t) * (k * t))));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(l / N[(t * N[(N[(k * t), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   5. lower-pow.f64 51.2%

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 51.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. associate-/l* N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   6. lower-/.f64 55.1%

      \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \color{blue}{{k}^{2}}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{\color{blue}{2}}} \]

   10. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot \color{blue}{k}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot \color{blue}{k}} \]

   13. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({t}^{3} \cdot k\right) \cdot k} \]

   15. unpow3 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   16. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   17. lower-*.f64 59.5%

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

6. Applied rewrites 59.5%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(t \cdot t\right) \cdot t\right) \cdot k\right) \cdot k} \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   5. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   7. associate-*l* N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

   9. lower-*.f64 58.1%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

8. Applied rewrites 58.1%

   \[\leadsto \ell \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(\color{blue}{k} \cdot k\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   4. unswap-sqr N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}\right)} \]

   6. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(\color{blue}{t} \cdot k\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

   9. lift-*.f64 65.9%

      \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{t}\right)\right)} \]

10. Applied rewrites 65.9%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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))))