Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.9% → 93.3%

Time: 8.2s

Alternatives: 21

Speedup: 4.7×

• 27.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{\frac{2}{t\_3 \cdot \tan k}}{t\_2 \cdot \left(\sin k \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}}{t\_2 \cdot \tan k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (fma (/ k (* t_m t_m)) k 2.0)) (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 1.75e-11)
      (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
      (if (<= t_m 6.2e+207)
        (/ (/ 2.0 (* t_3 (tan k))) (* t_2 (* (sin k) t_3)))
        (/
         (/ 2.0 (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l))))
         (* t_2 (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma((k / (t_m * t_m)), k, 2.0);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 1.75e-11) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 6.2e+207) {
		tmp = (2.0 / (t_3 * tan(k))) / (t_2 * (sin(k) * t_3));
	} else {
		tmp = (2.0 / ((sin(k) * t_m) * ((t_m / l) * (t_m / l)))) / (t_2 * tan(k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = fma(Float64(k / Float64(t_m * t_m)), k, 2.0)
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 1.75e-11)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 6.2e+207)
		tmp = Float64(Float64(2.0 / Float64(t_3 * tan(k))) / Float64(t_2 * Float64(sin(k) * t_3)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l)))) / Float64(t_2 * tan(k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.75e-11], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+207], N[(N[(2.0 / N[(t$95$3 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.7500000000000001e-11

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 1.7500000000000001e-11 < t < 6.2000000000000005e207

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.0

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{1.5}}{\ell} \cdot \tan k}}{\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}} \]

   if 6.2000000000000005e207 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 2: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t\_3 \cdot \left(t\_3 \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)))
        (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 1.75e-11)
      (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
      (if (<= t_m 6.2e+207)
        (/ 2.0 (* (sin k) (* t_3 (* t_3 t_2))))
        (/ (/ 2.0 (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l)))) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma((k / (t_m * t_m)), k, 2.0) * tan(k);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 1.75e-11) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 6.2e+207) {
		tmp = 2.0 / (sin(k) * (t_3 * (t_3 * t_2)));
	} else {
		tmp = (2.0 / ((sin(k) * t_m) * ((t_m / l) * (t_m / l)))) / t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k))
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 1.75e-11)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 6.2e+207)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(t_3 * Float64(t_3 * t_2))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l)))) / t_2);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.75e-11], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+207], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t$95$3 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.7500000000000001e-11

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 1.7500000000000001e-11 < t < 6.2000000000000005e207

   1. Initial program 53.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. unpow3 N/A

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

      7. lift-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 74.0

          \[\leadsto \frac{2}{\sin k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \tan k\right)\right)}\right)} \]

   5. Applied rewrites 64.9%

      \[\leadsto \frac{2}{\sin k \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \tan k\right)\right)\right)}} \]

   if 6.2000000000000005e207 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 3: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 12500000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\left(\sin k \cdot t\_3\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)))
        (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 12500000000000.0)
      (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
      (if (<= t_m 6.2e+207)
        (/ 2.0 (* t_3 (* (* (sin k) t_3) t_2)))
        (/ (/ 2.0 (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l)))) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma((k / (t_m * t_m)), k, 2.0) * tan(k);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 12500000000000.0) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 6.2e+207) {
		tmp = 2.0 / (t_3 * ((sin(k) * t_3) * t_2));
	} else {
		tmp = (2.0 / ((sin(k) * t_m) * ((t_m / l) * (t_m / l)))) / t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k))
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 12500000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 6.2e+207)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(sin(k) * t_3) * t_2)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l)))) / t_2);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 12500000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+207], N[(2.0 / N[(t$95$3 * N[(N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.25e13

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 1.25e13 < t < 6.2000000000000005e207

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 64.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{1.5}}{\ell} \cdot \left(\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \tan k\right)\right)}} \]

   if 6.2000000000000005e207 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 4: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(t\_3 \cdot \tan k\right)\right) \cdot \left(t\_3 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}}{t\_2 \cdot \tan k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (fma (/ k (* t_m t_m)) k 2.0)) (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 1.65e+25)
      (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
      (if (<= t_m 6.2e+207)
        (/ 2.0 (* (* (sin k) (* t_3 (tan k))) (* t_3 t_2)))
        (/
         (/ 2.0 (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l))))
         (* t_2 (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = fma((k / (t_m * t_m)), k, 2.0);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 1.65e+25) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 6.2e+207) {
		tmp = 2.0 / ((sin(k) * (t_3 * tan(k))) * (t_3 * t_2));
	} else {
		tmp = (2.0 / ((sin(k) * t_m) * ((t_m / l) * (t_m / l)))) / (t_2 * tan(k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = fma(Float64(k / Float64(t_m * t_m)), k, 2.0)
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 1.65e+25)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 6.2e+207)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(t_3 * tan(k))) * Float64(t_3 * t_2)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l)))) / Float64(t_2 * tan(k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e+25], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+207], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$3 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.6500000000000001e25

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 1.6500000000000001e25 < t < 6.2000000000000005e207

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.0

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 73.8

          \[\leadsto \frac{2}{\left(\sin k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \tan k\right)\right) \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      13. lift-+.f64 N/A

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

   7. Applied rewrites 64.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \tan k\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)\right)}} \]

   if 6.2000000000000005e207 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9200000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \frac{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot t\_m\right)}{\ell}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9200000000.0)
    (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
    (if (<= t_m 1.3e+194)
      (/
       2.0
       (*
        (*
         (fma (/ k (* t_m t_m)) k 2.0)
         (/ (* (/ t_m l) (* (* (sin k) t_m) t_m)) l))
        (tan k)))
      (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9200000000.0) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 1.3e+194) {
		tmp = 2.0 / ((fma((k / (t_m * t_m)), k, 2.0) * (((t_m / l) * ((sin(k) * t_m) * t_m)) / l)) * tan(k));
	} else {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9200000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 1.3e+194)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * Float64(Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * t_m)) / l)) * tan(k)));
	else
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9200000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+194], N[(2.0 / N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.2e9

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 9.2e9 < t < 1.2999999999999999e194

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.0

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 60.0

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

   9. Applied rewrites 60.0%

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

   if 1.2999999999999999e194 < t

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2900000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}}{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2900000000000.0)
    (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
    (/
     (/ 2.0 (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l))))
     (* (fma (/ k (* t_m t_m)) k 2.0) (tan k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2900000000000.0) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else {
		tmp = (2.0 / ((sin(k) * t_m) * ((t_m / l) * (t_m / l)))) / (fma((k / (t_m * t_m)), k, 2.0) * tan(k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2900000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l)))) / Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2900000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.9e12

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 2.9e12 < t

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 7: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2800000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \left(\left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2800000000000.0)
    (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
    (if (<= t_m 5.5e+125)
      (/
       2.0
       (*
        (*
         (fma (/ k (* t_m t_m)) k 2.0)
         (* (* (* t_m t_m) (/ t_m l)) (/ (sin k) l)))
        (tan k)))
      (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2800000000000.0) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 5.5e+125) {
		tmp = 2.0 / ((fma((k / (t_m * t_m)), k, 2.0) * (((t_m * t_m) * (t_m / l)) * (sin(k) / l))) * tan(k));
	} else {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2800000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 5.5e+125)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l)) * Float64(sin(k) / l))) * tan(k)));
	else
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2800000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+125], N[(2.0 / N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8e12

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 2.8e12 < t < 5.49999999999999996e125

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 69.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Applied rewrites 69.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.0

          \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. unpow3 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. unpow3 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 57.7

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

   9. Applied rewrites 57.7%

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

   if 5.49999999999999996e125 < t

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2800000000000:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\sin k \cdot \frac{\left(\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2800000000000.0)
    (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l))))
    (if (<= t_m 5.5e+125)
      (/
       2.0
       (*
        (sin k)
        (/
         (*
          (* (fma (/ k (* t_m t_m)) k 2.0) (tan k))
          (* (* t_m t_m) (/ t_m l)))
         l)))
      (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2800000000000.0) {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	} else if (t_m <= 5.5e+125) {
		tmp = 2.0 / (sin(k) * (((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) * ((t_m * t_m) * (t_m / l))) / l));
	} else {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2800000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	elseif (t_m <= 5.5e+125)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) * Float64(Float64(t_m * t_m) * Float64(t_m / l))) / l)));
	else
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2800000000000.0], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+125], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8e12

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

   if 2.8e12 < t < 5.49999999999999996e125

   1. Initial program 53.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow3 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

   3. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. cube-mult N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 58.1%

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

   if 5.49999999999999996e125 < t

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{t\_m \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e+51)
    (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))
    (/ 2.0 (* (/ k l) (* (* (sin k) (tan k)) (/ (* t_m k) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.2e+51) {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	} else {
		tmp = 2.0 / ((k / l) * ((sin(k) * tan(k)) * ((t_m * k) / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.2e+51)
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.2e+51], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.2000000000000002e51

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

   if 9.2000000000000002e51 < k

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 58.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

   8. Applied rewrites 73.1%

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

Alternative 10: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \left(k \cdot \frac{k}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e+51)
    (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))
    (/ 2.0 (* (* t_m (* (tan k) (sin k))) (* k (/ k (* l l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.2e+51) {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	} else {
		tmp = 2.0 / ((t_m * (tan(k) * sin(k))) * (k * (k / (l * l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.2e+51)
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(tan(k) * sin(k))) * Float64(k * Float64(k / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.2e+51], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.2000000000000002e51

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

   if 9.2000000000000002e51 < k

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 62.1%

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

Alternative 11: 70.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e+51)
    (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))
    (/ 2.0 (* (tan k) (* (sin k) (* (* (/ t_m (* l l)) k) k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.2e+51) {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	} else {
		tmp = 2.0 / (tan(k) * (sin(k) * (((t_m / (l * l)) * k) * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.2e+51)
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.2e+51], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.2000000000000002e51

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

   if 9.2000000000000002e51 < k

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.7

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

   8. Applied rewrites 66.7%

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

Alternative 12: 68.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+51}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e+51)
    (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0))))
    (/ 2.0 (* k (* k (* (* (tan k) (sin k)) (/ t_m (* l l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.2e+51) {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	} else {
		tmp = 2.0 / (k * (k * ((tan(k) * sin(k)) * (t_m / (l * l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.2e+51)
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * sin(k)) * Float64(t_m / Float64(l * l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.2e+51], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.2000000000000002e51

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

   if 9.2000000000000002e51 < k

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 63.9%

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

Alternative 13: 68.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-56)
    (/ 2.0 (* (/ (* (* (* k k) k) k) l) (/ t_m l)))
    (exp (- (* (log l) 2.0) (fma (log t_m) 3.0 (* (log k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-56) {
		tmp = 2.0 / (((((k * k) * k) * k) / l) * (t_m / l));
	} else {
		tmp = exp(((log(l) * 2.0) - fma(log(t_m), 3.0, (log(k) * 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.8e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * k) * k) / l) * Float64(t_m / l)));
	else
		tmp = exp(Float64(Float64(log(l) * 2.0) - fma(log(t_m), 3.0, Float64(log(k) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-56], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.8e-56

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   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 51.4

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

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   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. lift-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 56.0

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

   9. Applied rewrites 56.0%

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

   if 7.8e-56 < t

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 58.3

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. pow2 N/A

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

      5. pow-to-exp N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lift-log.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. cube-mult N/A

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

      16. pow-to-exp N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      17. lift-log.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2} \cdot e^{\log t \cdot 3}} \]

      18. exp-sum N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log k \cdot 2 + \log t \cdot 3}} \]

      19. +-commutative N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      20. lift-fma.f64 N/A

          \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e-56)
    (/ 2.0 (* (/ (* (* (* k k) k) k) l) (/ t_m l)))
    (if (<= t_m 4e+71)
      (* (/ l (* (* (* t_m t_m) k) t_m)) (/ l k))
      (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e-56) {
		tmp = 2.0 / (((((k * k) * k) * k) / l) * (t_m / l));
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.9d-56) then
        tmp = 2.0d0 / (((((k * k) * k) * k) / l) * (t_m / l))
    else if (t_m <= 4d+71) then
        tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
    else
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e-56) {
		tmp = 2.0 / (((((k * k) * k) * k) / l) * (t_m / l));
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.9e-56:
		tmp = 2.0 / (((((k * k) * k) * k) / l) * (t_m / l))
	elif t_m <= 4e+71:
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
	else:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.9e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * k) * k) / l) * Float64(t_m / l)));
	elseif (t_m <= 4e+71)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * t_m) * k) * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.9e-56)
		tmp = 2.0 / (((((k * k) * k) * k) / l) * (t_m / l));
	elseif (t_m <= 4e+71)
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	else
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e-56], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+71], N[(N[(l / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.89999999999999991e-56

   1. Initial program 53.9%

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

   2. Taylor expanded in t around 0

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

      1. 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 59.7

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

   4. Applied rewrites 59.7%

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

   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 51.4

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

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   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. lift-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 56.0

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

   9. Applied rewrites 56.0%

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

   if 2.89999999999999991e-56 < t < 4.0000000000000002e71

   1. Initial program 53.9%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

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

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

      7. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{k} \]

      11. lower-/.f64 63.5

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k}} \]

   10. Applied rewrites 63.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 4.0000000000000002e71 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 68.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{t\_m \cdot \frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-55)
    (/ 2.0 (* t_m (/ (* (* (* k k) k) k) (* l l))))
    (if (<= t_m 4e+71)
      (* (/ l (* (* (* t_m t_m) k) t_m)) (/ l k))
      (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-55) {
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l * l)));
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4d-55) then
        tmp = 2.0d0 / (t_m * ((((k * k) * k) * k) / (l * l)))
    else if (t_m <= 4d+71) then
        tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
    else
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-55) {
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l * l)));
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4e-55:
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l * l)))
	elif t_m <= 4e+71:
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
	else:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e-55)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(Float64(k * k) * k) * k) / Float64(l * l))));
	elseif (t_m <= 4e+71)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * t_m) * k) * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4e-55)
		tmp = 2.0 / (t_m * ((((k * k) * k) * k) / (l * l)));
	elseif (t_m <= 4e+71)
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	else
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-55], N[(2.0 / N[(t$95$m * N[(N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+71], N[(N[(l / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.99999999999999998e-55

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. 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 59.7

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

   4. Applied rewrites 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   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 51.4

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   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 51.5

         \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\ell \cdot \color{blue}{\ell}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{\ell \cdot \ell}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{\left(2 \cdot 2\right)}}{\ell \cdot \ell}} \]

      12. pow-sqr N/A

          \[\leadsto \frac{2}{t \cdot \frac{{k}^{2} \cdot {k}^{2}}{\ell \cdot \ell}} \]

      13. pow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{\left(k \cdot k\right) \cdot {k}^{2}}{\ell \cdot \ell}} \]

      14. pow2 N/A

          \[\leadsto \frac{2}{t \cdot \frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{t \cdot \frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell \cdot \ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell \cdot \ell}} \]

      18. lower-*.f64 51.5

          \[\leadsto \frac{2}{t \cdot \frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell \cdot \ell}} \]

   9. Applied rewrites 51.5%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\frac{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}{\ell \cdot \ell}}} \]

   if 3.99999999999999998e-55 < t < 4.0000000000000002e71

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

      7. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{k} \]

      11. lower-/.f64 63.5

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k}} \]

   10. Applied rewrites 63.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 4.0000000000000002e71 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-106)
    (/ (/ (* l l) (* (* (* k k) t_m) t_m)) t_m)
    (if (<= t_m 4e+71)
      (* (/ l (* (* (* t_m t_m) k) t_m)) (/ l k))
      (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-106) {
		tmp = ((l * l) / (((k * k) * t_m) * t_m)) / t_m;
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d-106) then
        tmp = ((l * l) / (((k * k) * t_m) * t_m)) / t_m
    else if (t_m <= 4d+71) then
        tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
    else
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-106) {
		tmp = ((l * l) / (((k * k) * t_m) * t_m)) / t_m;
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8e-106:
		tmp = ((l * l) / (((k * k) * t_m) * t_m)) / t_m
	elif t_m <= 4e+71:
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
	else:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-106)
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(Float64(k * k) * t_m) * t_m)) / t_m);
	elseif (t_m <= 4e+71)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * t_m) * k) * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e-106)
		tmp = ((l * l) / (((k * k) * t_m) * t_m)) / t_m;
	elseif (t_m <= 4e+71)
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	else
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-106], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 4e+71], N[(N[(l / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.99999999999999953e-106

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(t \cdot t\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t} \]

      11. lower-*.f64 58.8

          \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{t} \]

   8. Applied rewrites 58.8%

      \[\leadsto \frac{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}}{\color{blue}{t}} \]

   if 7.99999999999999953e-106 < t < 4.0000000000000002e71

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

      7. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{k} \]

      11. lower-/.f64 63.5

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k}} \]

   10. Applied rewrites 63.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 4.0000000000000002e71 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 39.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-58}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)\right)} \cdot \ell\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6e-58)
    (* (/ l (* t_m (* t_m (* (* k k) t_m)))) l)
    (if (<= t_m 4e+71)
      (* (/ l (* (* (* t_m t_m) k) t_m)) (/ l k))
      (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6e-58) {
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l;
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6d-58) then
        tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l
    else if (t_m <= 4d+71) then
        tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
    else
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6e-58) {
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l;
	} else if (t_m <= 4e+71) {
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6e-58:
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l
	elif t_m <= 4e+71:
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k)
	else:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6e-58)
		tmp = Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(Float64(k * k) * t_m)))) * l);
	elseif (t_m <= 4e+71)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * t_m) * k) * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6e-58)
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l;
	elseif (t_m <= 4e+71)
		tmp = (l / (((t_m * t_m) * k) * t_m)) * (l / k);
	else
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6e-58], N[(N[(l / N[(t$95$m * N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[t$95$m, 4e+71], N[(N[(l / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.00000000000000015e-58

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 62.0

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   10. Applied rewrites 62.0%

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   if 6.00000000000000015e-58 < t < 4.0000000000000002e71

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot \color{blue}{k}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right) \cdot \color{blue}{k}} \]

      7. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{k} \]

      11. lower-/.f64 63.5

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \frac{\ell}{\color{blue}{k}} \]

   10. Applied rewrites 63.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{k}} \]

   if 4.0000000000000002e71 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 30.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-160}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3e-160)
    (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l)
    (* l (/ (/ (/ l (* (* k k) t_m)) t_m) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3e-160) {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	} else {
		tmp = l * (((l / ((k * k) * t_m)) / t_m) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3d-160) then
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    else
        tmp = l * (((l / ((k * k) * t_m)) / t_m) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3e-160) {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	} else {
		tmp = l * (((l / ((k * k) * t_m)) / t_m) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3e-160:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	else:
		tmp = l * (((l / ((k * k) * t_m)) / t_m) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3e-160)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	else
		tmp = Float64(l * Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) / t_m) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3e-160)
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	else
		tmp = l * (((l / ((k * k) * t_m)) / t_m) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3e-160], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(l * N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999997e-160

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   if 2.99999999999999997e-160 < k

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{\color{blue}{t \cdot t}} \]

      4. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t \cdot \color{blue}{t}} \]

      5. associate-/r* N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]

      6. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]

      7. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{t} \]

      8. lower-/.f64 63.5

         \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{t} \]

   8. Applied rewrites 63.5%

      \[\leadsto \ell \cdot \frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 27.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.6e-160)
    (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l)
    (* (/ l (* (* (* k k) t_m) t_m)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-160) {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	} else {
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.6d-160) then
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    else
        tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-160) {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	} else {
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.6e-160:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	else:
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.6e-160)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.6e-160)
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	else
		tmp = (l / (((k * k) * t_m) * t_m)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.6e-160], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.60000000000000003e-160

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   if 2.60000000000000003e-160 < k

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      7. times-frac N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{t} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{t} \]

      11. lower-/.f64 63.4

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{\color{blue}{t}} \]

   8. Applied rewrites 63.4%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 27.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot k\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-106)
    (* (/ l (* t_m (* t_m (* (* k k) t_m)))) l)
    (* (/ l (* (* (* t_m k) t_m) (* t_m k))) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-106) {
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l;
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.4d-106) then
        tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l
    else
        tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-106) {
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l;
	} else {
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.4e-106:
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l
	else:
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.4e-106)
		tmp = Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(Float64(k * k) * t_m)))) * l);
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t_m * k) * t_m) * Float64(t_m * k))) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.4e-106)
		tmp = (l / (t_m * (t_m * ((k * k) * t_m)))) * l;
	else
		tmp = (l / (((t_m * k) * t_m) * (t_m * k))) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-106], N[(N[(l / N[(t$95$m * N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3999999999999998e-106

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      9. associate-*l* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 62.0

          \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   10. Applied rewrites 62.0%

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   if 2.3999999999999998e-106 < t

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.6

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. cube-mult N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      13. lower-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

      15. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

      16. lift-*.f64 58.3

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      3. lower-*.f64 58.3

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      13. lower-*.f64 63.0

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   8. Applied rewrites 63.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

      6. lower-*.f64 66.5

         \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 27.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)\right)} \cdot \ell\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* t_m (* t_m (* (* k k) t_m)))) l)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * (t_m * ((k * k) * t_m)))) * l);
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (t_m * (t_m * ((k * k) * t_m)))) * l)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / (t_m * (t_m * ((k * k) * t_m)))) * l);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / (t_m * (t_m * ((k * k) * t_m)))) * l)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(Float64(k * k) * t_m)))) * l))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (t_m * (t_m * ((k * k) * t_m)))) * l);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(t$95$m * N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 50.6

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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. lift-pow.f64 N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   9. cube-mult N/A

      \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   11. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   13. lower-*.f64 58.3

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left({k}^{2} \cdot t\right) \cdot \left(t \cdot t\right)} \]

   15. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

   16. lift-*.f64 58.3

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \]

6. Applied rewrites 58.3%

   \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   3. lower-*.f64 58.3

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   13. lower-*.f64 63.0

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

8. Applied rewrites 63.0%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   7. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   9. associate-*l* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 62.0

       \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]

10. Applied rewrites 62.0%

    \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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))))