Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 80.6%

Time: 8.1s

Alternatives: 19

Speedup: 6.6×

• 27.3% 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: 54.8% 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: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.85 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot t\_m}{{l\_m}^{2}} \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m}{l\_m} \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.85e-71)
    (/ 2.0 (* (/ (* (pow k_m 2.0) t_m) (pow l_m 2.0)) (* (tan k_m) (sin k_m))))
    (if (<= t_m 4e+132)
      (/
       2.0
       (*
        (* (/ (* (* (* t_m t_m) (/ t_m l_m)) (sin k_m)) l_m) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.85e-71) {
		tmp = 2.0 / (((pow(k_m, 2.0) * t_m) / pow(l_m, 2.0)) * (tan(k_m) * sin(k_m)));
	} else if (t_m <= 4e+132) {
		tmp = 2.0 / ((((((t_m * t_m) * (t_m / l_m)) * sin(k_m)) / l_m) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.85e-71)
		tmp = Float64(2.0 / Float64(Float64(Float64((k_m ^ 2.0) * t_m) / (l_m ^ 2.0)) * Float64(tan(k_m) * sin(k_m))));
	elseif (t_m <= 4e+132)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * Float64(t_m / l_m)) * sin(k_m)) / l_m) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.85e-71], N[(2.0 / N[(N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+132], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8500000000000001e-71

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 54.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 61.4

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

   6. Applied rewrites 61.4%

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

   if 2.8500000000000001e-71 < t < 3.99999999999999996e132

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 66.1

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

   3. Applied rewrites 66.1%

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

   if 3.99999999999999996e132 < t

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.85 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot t\_m}{{l\_m}^{2}} \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.85e-71)
    (/ 2.0 (* (/ (* (pow k_m 2.0) t_m) (pow l_m 2.0)) (* (tan k_m) (sin k_m))))
    (if (<= t_m 5e+142)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l_m) (/ t_m l_m)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.85e-71) {
		tmp = 2.0 / (((pow(k_m, 2.0) * t_m) / pow(l_m, 2.0)) * (tan(k_m) * sin(k_m)));
	} else if (t_m <= 5e+142) {
		tmp = 2.0 / ((((((t_m * t_m) / l_m) * (t_m / l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.85e-71)
		tmp = Float64(2.0 / Float64(Float64(Float64((k_m ^ 2.0) * t_m) / (l_m ^ 2.0)) * Float64(tan(k_m) * sin(k_m))));
	elseif (t_m <= 5e+142)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(t_m / l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.85e-71], N[(2.0 / N[(N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+142], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8500000000000001e-71

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 54.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 61.4

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

   6. Applied rewrites 61.4%

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

   if 2.8500000000000001e-71 < t < 5.0000000000000001e142

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 65.4

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

   3. Applied rewrites 65.4%

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

   if 5.0000000000000001e142 < t

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

Alternative 3: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.118:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot t\_m}{{l\_m}^{2}} \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.118)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (/
     2.0
     (* (/ (* (pow k_m 2.0) t_m) (pow l_m 2.0)) (* (tan k_m) (sin k_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.118) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else {
		tmp = 2.0 / (((pow(k_m, 2.0) * t_m) / pow(l_m, 2.0)) * (tan(k_m) * sin(k_m)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.118)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k_m ^ 2.0) * t_m) / (l_m ^ 2.0)) * Float64(tan(k_m) * sin(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.118], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.11799999999999999

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 0.11799999999999999 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 54.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 61.4

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

   6. Applied rewrites 61.4%

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

Alternative 4: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.118:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot {k\_m}^{2}\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.118)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (/
     2.0
     (* (* (* (tan k_m) (sin k_m)) (pow k_m 2.0)) (/ t_m (* l_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.118) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else {
		tmp = 2.0 / (((tan(k_m) * sin(k_m)) * pow(k_m, 2.0)) * (t_m / (l_m * l_m)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.118)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k_m) * sin(k_m)) * (k_m ^ 2.0)) * Float64(t_m / Float64(l_m * l_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.118], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.11799999999999999

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 0.11799999999999999 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 68.8

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

   3. Applied rewrites 68.8%

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-pow.f64 59.1

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

   8. Applied rewrites 59.1%

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

Alternative 5: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-39}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{elif}\;k\_m \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\left(\left(\sin k\_m \cdot t\_m\right) \cdot \left(\frac{t\_m}{l\_m \cdot l\_m} \cdot t\_m\right)\right) \cdot \tan k\_m}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.2e-39)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (if (<= k_m 3.5e+148)
      (/ 1.0 (* (* (* (sin k_m) t_m) (* (/ t_m (* l_m l_m)) t_m)) (tan k_m)))
      (* l_m (/ l_m (* (* (* (* k_m k_m) t_m) t_m) t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 3.2e-39) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else if (k_m <= 3.5e+148) {
		tmp = 1.0 / (((sin(k_m) * t_m) * ((t_m / (l_m * l_m)) * t_m)) * tan(k_m));
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-39)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	elseif (k_m <= 3.5e+148)
		tmp = Float64(1.0 / Float64(Float64(Float64(sin(k_m) * t_m) * Float64(Float64(t_m / Float64(l_m * l_m)) * t_m)) * tan(k_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.2e-39], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k$95$m, 3.5e+148], N[(1.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.1999999999999998e-39

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 3.1999999999999998e-39 < k < 3.4999999999999999e148

   1. Initial program 54.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 50.6

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 61.6%

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

   if 3.4999999999999999e148 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

Alternative 6: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-32}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{elif}\;k\_m \leq 3.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\left(\tan k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \frac{t\_m}{l\_m \cdot l\_m}\right)\right) \cdot \sin k\_m}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2e-32)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (if (<= k_m 3.5e+148)
      (/ 1.0 (* (* (tan k_m) (* (* t_m t_m) (/ t_m (* l_m l_m)))) (sin k_m)))
      (* l_m (/ l_m (* (* (* (* k_m k_m) t_m) t_m) t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 2e-32) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else if (k_m <= 3.5e+148) {
		tmp = 1.0 / ((tan(k_m) * ((t_m * t_m) * (t_m / (l_m * l_m)))) * sin(k_m));
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 2e-32)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	elseif (k_m <= 3.5e+148)
		tmp = Float64(1.0 / Float64(Float64(tan(k_m) * Float64(Float64(t_m * t_m) * Float64(t_m / Float64(l_m * l_m)))) * sin(k_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2e-32], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k$95$m, 3.5e+148], N[(1.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.00000000000000011e-32

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 2.00000000000000011e-32 < k < 3.4999999999999999e148

   1. Initial program 54.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 50.6

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 50.6

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

   6. Applied rewrites 50.6%

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

   8. Applied rewrites 56.8%

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

   if 3.4999999999999999e148 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

Alternative 7: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-66}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(l\_m \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\right) \cdot \cos k\_m}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-66)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (if (<= k_m 1.35e+150)
      (/ (* (* l_m (/ l_m (* (* t_m t_m) t_m))) (cos k_m)) (* k_m k_m))
      (* l_m (/ l_m (* (* (* (* k_m k_m) t_m) t_m) t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 2.5e-66) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else if (k_m <= 1.35e+150) {
		tmp = ((l_m * (l_m / ((t_m * t_m) * t_m))) * cos(k_m)) / (k_m * k_m);
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-66)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	elseif (k_m <= 1.35e+150)
		tmp = Float64(Float64(Float64(l_m * Float64(l_m / Float64(Float64(t_m * t_m) * t_m))) * cos(k_m)) / Float64(k_m * k_m));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-66], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k$95$m, 1.35e+150], N[(N[(N[(l$95$m * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.49999999999999981e-66

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 2.49999999999999981e-66 < k < 1.35000000000000004e150

   1. Initial program 54.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 50.6

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 56.5

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 56.5

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if 1.35000000000000004e150 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

Alternative 8: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-66}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{elif}\;k\_m \leq 7 \cdot 10^{+149}:\\ \;\;\;\;\frac{l\_m}{k\_m \cdot k\_m} \cdot \left(\cos k\_m \cdot \frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-66)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (if (<= k_m 7e+149)
      (* (/ l_m (* k_m k_m)) (* (cos k_m) (/ l_m (* (* t_m t_m) t_m))))
      (* l_m (/ l_m (* (* (* (* k_m k_m) t_m) t_m) t_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 2.5e-66) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else if (k_m <= 7e+149) {
		tmp = (l_m / (k_m * k_m)) * (cos(k_m) * (l_m / ((t_m * t_m) * t_m)));
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-66)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	elseif (k_m <= 7e+149)
		tmp = Float64(Float64(l_m / Float64(k_m * k_m)) * Float64(cos(k_m) * Float64(l_m / Float64(Float64(t_m * t_m) * t_m))));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-66], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k$95$m, 7e+149], N[(N[(l$95$m / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.49999999999999981e-66

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 2.49999999999999981e-66 < k < 7.00000000000000023e149

   1. Initial program 54.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 50.6

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 57.0

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

      20. lift-pow.f64 N/A

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

      21. unpow3 N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 57.0

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

   9. Applied rewrites 57.0%

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

   if 7.00000000000000023e149 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

Alternative 9: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{l\_m}^{2} \cdot \cos k\_m}{e^{\mathsf{fma}\left(\log k\_m, 2, \log t\_m \cdot 3\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 27.0)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (/
     (* (pow l_m 2.0) (cos k_m))
     (exp (fma (log k_m) 2.0 (* (log t_m) 3.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 27.0) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else {
		tmp = (pow(l_m, 2.0) * cos(k_m)) / exp(fma(log(k_m), 2.0, (log(t_m) * 3.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 27.0)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	else
		tmp = Float64(Float64((l_m ^ 2.0) * cos(k_m)) / exp(fma(log(k_m), 2.0, Float64(log(t_m) * 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 27.0], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(N[Log[k$95$m], $MachinePrecision] * 2.0 + N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 27

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 27 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 50.6

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{t}^{3} \cdot {\sin k}^{2}} \]

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-log.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 62.8

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

   9. Applied rewrites 62.8%

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

Alternative 10: 71.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.65 \cdot 10^{-135}:\\ \;\;\;\;e^{\log l\_m \cdot 2 - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{l\_m}{\left(k\_m \cdot t\_m\right) \cdot k\_m}}{t\_m}}{t\_m} \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.65e-135)
    (exp (- (* (log l_m) 2.0) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (* (/ (/ (/ l_m (* (* k_m t_m) k_m)) t_m) t_m) l_m))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.65e-135) {
		tmp = exp(((log(l_m) * 2.0) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else {
		tmp = (((l_m / ((k_m * t_m) * k_m)) / t_m) / t_m) * l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.65e-135)
		tmp = exp(Float64(Float64(log(l_m) * 2.0) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(l_m / Float64(Float64(k_m * t_m) * k_m)) / t_m) / t_m) * l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.65e-135], N[Exp[N[(N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(l$95$m / N[(N[(k$95$m * t$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.65e-135

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

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

   if 1.65e-135 < k

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.6

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

   8. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* 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. 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. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 62.6

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 66.1

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

   10. Applied rewrites 66.1%

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

Alternative 11: 68.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\frac{l\_m}{\left(k\_m \cdot t\_m\right) \cdot k\_m}}{t\_m}}{t\_m} \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 10^{+169}:\\ \;\;\;\;\frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot k\_m} \cdot \frac{l\_m}{k\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (/ (/ (/ l_m (* (* k_m t_m) k_m)) t_m) t_m) l_m)))
   (*
    t_s
    (if (<= t_m 9.2e-46)
      t_2
      (if (<= t_m 1e+169)
        (* (/ l_m (* (* t_m t_m) k_m)) (/ l_m (* k_m t_m)))
        t_2)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = (((l_m / ((k_m * t_m) * k_m)) / t_m) / t_m) * l_m;
	double tmp;
	if (t_m <= 9.2e-46) {
		tmp = t_2;
	} else if (t_m <= 1e+169) {
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (((l_m / ((k_m * t_m) * k_m)) / t_m) / t_m) * l_m
    if (t_m <= 9.2d-46) then
        tmp = t_2
    else if (t_m <= 1d+169) then
        tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m))
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = (((l_m / ((k_m * t_m) * k_m)) / t_m) / t_m) * l_m;
	double tmp;
	if (t_m <= 9.2e-46) {
		tmp = t_2;
	} else if (t_m <= 1e+169) {
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m));
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	t_2 = (((l_m / ((k_m * t_m) * k_m)) / t_m) / t_m) * l_m
	tmp = 0
	if t_m <= 9.2e-46:
		tmp = t_2
	elif t_m <= 1e+169:
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m))
	else:
		tmp = t_2
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(Float64(Float64(Float64(l_m / Float64(Float64(k_m * t_m) * k_m)) / t_m) / t_m) * l_m)
	tmp = 0.0
	if (t_m <= 9.2e-46)
		tmp = t_2;
	elseif (t_m <= 1e+169)
		tmp = Float64(Float64(l_m / Float64(Float64(t_m * t_m) * k_m)) * Float64(l_m / Float64(k_m * t_m)));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	t_2 = (((l_m / ((k_m * t_m) * k_m)) / t_m) / t_m) * l_m;
	tmp = 0.0;
	if (t_m <= 9.2e-46)
		tmp = t_2;
	elseif (t_m <= 1e+169)
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m));
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[(N[(N[(l$95$m / N[(N[(k$95$m * t$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.2e-46], t$95$2, If[LessEqual[t$95$m, 1e+169], N[(N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.1999999999999997e-46 or 9.99999999999999934e168 < t

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.7

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

      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. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 62.6

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

   8. Applied rewrites 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* 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. 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. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 62.6

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 66.1

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

   10. Applied rewrites 66.1%

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

   if 9.1999999999999997e-46 < t < 9.99999999999999934e168

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

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

   4. Applied rewrites 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 55.2

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. cube-mult N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 57.7

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

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

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

      6. pow-to-exp N/A

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

      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 69.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. exp-diff N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\color{blue}{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. exp-sum N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot \color{blue}{e^{\log k \cdot 2}}} \]

      10. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot e^{\log \color{blue}{k} \cdot 2}} \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot e^{\color{blue}{\log k \cdot 2}}} \]

      12. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\color{blue}{\log k \cdot 2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\color{blue}{\log k} \cdot 2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\log k \cdot 2}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\log k \cdot 2}} \]

      16. pow-to-exp N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      19. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

      20. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left({k}^{2} \cdot t\right)} \]

      21. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 68.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-166}:\\ \;\;\;\;e^{\log \left(l\_m \cdot l\_m\right) - \mathsf{fma}\left(\log t\_m, 3, \log k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.5e-166)
    (exp (- (log (* l_m l_m)) (fma (log t_m) 3.0 (* (log k_m) 2.0))))
    (/ l_m (* (* (* (* k_m k_m) t_m) t_m) (/ t_m l_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.5e-166) {
		tmp = exp((log((l_m * l_m)) - fma(log(t_m), 3.0, (log(k_m) * 2.0))));
	} else {
		tmp = l_m / ((((k_m * k_m) * t_m) * t_m) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-166)
		tmp = exp(Float64(log(Float64(l_m * l_m)) - fma(log(t_m), 3.0, Float64(log(k_m) * 2.0))));
	else
		tmp = Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.5e-166], N[Exp[N[(N[Log[N[(l$95$m * l$95$m), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(N[Log[k$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.5000000000000001e-166

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(t \cdot t\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{e^{\log \ell \cdot 2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(t \cdot t\right)} \]

      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 69.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      2. *-commutative N/A

         \[\leadsto e^{2 \cdot \log \ell - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto e^{2 \cdot \log \ell - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      4. log-pow-rev N/A

         \[\leadsto e^{\log \left({\ell}^{2}\right) - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      5. pow2 N/A

         \[\leadsto e^{\log \left(\ell \cdot \ell\right) - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto e^{\log \left(\ell \cdot \ell\right) - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      7. lower-log.f64 63.8

         \[\leadsto e^{\log \left(\ell \cdot \ell\right) - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

   10. Applied rewrites 63.8%

       \[\leadsto e^{\log \left(\ell \cdot \ell\right) - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

   if 1.5000000000000001e-166 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. div-flip N/A

         \[\leadsto \ell \cdot \frac{1}{\color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \frac{\ell}{\color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\ell}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t}{\ell}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \frac{\color{blue}{t}}{\ell}} \]

      12. lower-/.f64 62.3

          \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \frac{t}{\color{blue}{\ell}}} \]

   8. Applied rewrites 62.3%

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{l\_m}{t\_m \cdot \left(\left(k\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.75e-163)
    (* (/ l_m (* t_m (* (* k_m t_m) (* k_m t_m)))) l_m)
    (/ l_m (* (* (* (* k_m k_m) t_m) t_m) (/ t_m l_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.75e-163) {
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m;
	} else {
		tmp = l_m / ((((k_m * k_m) * t_m) * t_m) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.75d-163) then
        tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m
    else
        tmp = l_m / ((((k_m * k_m) * t_m) * t_m) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.75e-163) {
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m;
	} else {
		tmp = l_m / ((((k_m * k_m) * t_m) * t_m) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 1.75e-163:
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m
	else:
		tmp = l_m / ((((k_m * k_m) * t_m) * t_m) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.75e-163)
		tmp = Float64(Float64(l_m / Float64(t_m * Float64(Float64(k_m * t_m) * Float64(k_m * t_m)))) * l_m);
	else
		tmp = Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 1.75e-163)
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m;
	else
		tmp = l_m / ((((k_m * k_m) * t_m) * t_m) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.75e-163], N[(N[(l$95$m / N[(t$95$m * N[(N[(k$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.75000000000000014e-163

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.7

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.6

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. lower-*.f64 65.5

         \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   10. Applied rewrites 65.5%

       \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   if 1.75000000000000014e-163 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. div-flip N/A

         \[\leadsto \ell \cdot \frac{1}{\color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}}} \]

      4. mult-flip-rev N/A

         \[\leadsto \frac{\ell}{\color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}{\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\ell}{\frac{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t}{\ell}} \]

      9. associate-/l* N/A

         \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \frac{\color{blue}{t}}{\ell}} \]

      12. lower-/.f64 62.3

          \[\leadsto \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \frac{t}{\color{blue}{\ell}}} \]

   8. Applied rewrites 62.3%

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{l\_m}{t\_m \cdot \left(\left(k\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m} \cdot \frac{l\_m}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.75e-163)
    (* (/ l_m (* t_m (* (* k_m t_m) (* k_m t_m)))) l_m)
    (* (/ l_m (* (* (* k_m k_m) t_m) t_m)) (/ l_m t_m)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.75e-163) {
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m;
	} else {
		tmp = (l_m / (((k_m * k_m) * t_m) * t_m)) * (l_m / t_m);
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.75d-163) then
        tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m
    else
        tmp = (l_m / (((k_m * k_m) * t_m) * t_m)) * (l_m / t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.75e-163) {
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m;
	} else {
		tmp = (l_m / (((k_m * k_m) * t_m) * t_m)) * (l_m / t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 1.75e-163:
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m
	else:
		tmp = (l_m / (((k_m * k_m) * t_m) * t_m)) * (l_m / t_m)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.75e-163)
		tmp = Float64(Float64(l_m / Float64(t_m * Float64(Float64(k_m * t_m) * Float64(k_m * t_m)))) * l_m);
	else
		tmp = Float64(Float64(l_m / Float64(Float64(Float64(k_m * k_m) * t_m) * t_m)) * Float64(l_m / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 1.75e-163)
		tmp = (l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m;
	else
		tmp = (l_m / (((k_m * k_m) * t_m) * t_m)) * (l_m / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.75e-163], N[(N[(l$95$m / N[(t$95$m * N[(N[(k$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(N[(l$95$m / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.75000000000000014e-163

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.7

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.6

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

      9. lower-*.f64 65.5

         \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   10. Applied rewrites 65.5%

       \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   if 1.75000000000000014e-163 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 62.5

          \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \frac{\ell}{\color{blue}{t}} \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{l\_m}{\left(t\_m \cdot t\_m\right) \cdot k\_m} \cdot \frac{l\_m}{k\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.2e-27)
    (* (/ l_m (* (* t_m t_m) k_m)) (/ l_m (* k_m t_m)))
    (* l_m (/ l_m (* (* (* (* k_m k_m) t_m) t_m) t_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 5.2e-27) {
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m));
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-27) then
        tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m))
    else
        tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 5.2e-27) {
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m));
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 5.2e-27:
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m))
	else:
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-27)
		tmp = Float64(Float64(l_m / Float64(Float64(t_m * t_m) * k_m)) * Float64(l_m / Float64(k_m * t_m)));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-27)
		tmp = (l_m / ((t_m * t_m) * k_m)) * (l_m / (k_m * t_m));
	else
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.2e-27], N[(N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.20000000000000034e-27

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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. pow2 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(t \cdot t\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{e^{\log \ell \cdot 2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \left(t \cdot t\right)} \]

      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 69.7%

      \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      2. lift--.f64 N/A

         \[\leadsto e^{\log \ell \cdot 2 - \mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)} \]

      3. exp-diff N/A

         \[\leadsto \frac{e^{\log \ell \cdot 2}}{\color{blue}{e^{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\color{blue}{\log t}, 3, \log k \cdot 2\right)}} \]

      5. lift-log.f64 N/A

         \[\leadsto \frac{e^{\log \ell \cdot 2}}{e^{\mathsf{fma}\left(\log \color{blue}{t}, 3, \log k \cdot 2\right)}} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{{\ell}^{2}}{e^{\color{blue}{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\color{blue}{\mathsf{fma}\left(\log t, 3, \log k \cdot 2\right)}}} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3 + \log k \cdot 2}} \]

      9. exp-sum N/A

         \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot \color{blue}{e^{\log k \cdot 2}}} \]

      10. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{e^{\log t \cdot 3} \cdot e^{\log \color{blue}{k} \cdot 2}} \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\ell \cdot \ell}{{t}^{3} \cdot e^{\color{blue}{\log k \cdot 2}}} \]

      12. unpow3 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\color{blue}{\log k \cdot 2}}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\color{blue}{\log k} \cdot 2}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\log k \cdot 2}} \]

      15. lift-log.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot e^{\log k \cdot 2}} \]

      16. pow-to-exp N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      19. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left({k}^{2} \cdot \color{blue}{t}\right)} \]

      20. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left({k}^{2} \cdot t\right)} \]

      21. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 5.20000000000000034e-27 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. lower-*.f64 61.2

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t} \]

   8. Applied rewrites 61.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{l\_m}{\left(\left(k\_m \cdot t\_m\right) \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(\left(\left(k\_m \cdot k\_m\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.1e-76)
    (* (/ l_m (* (* (* k_m t_m) t_m) (* k_m t_m))) l_m)
    (* l_m (/ l_m (* (* (* (* k_m k_m) t_m) t_m) t_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.1e-76) {
		tmp = (l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m;
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.1d-76) then
        tmp = (l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m
    else
        tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.1e-76) {
		tmp = (l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m;
	} else {
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 4.1e-76:
		tmp = (l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m
	else:
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 4.1e-76)
		tmp = Float64(Float64(l_m / Float64(Float64(Float64(k_m * t_m) * t_m) * Float64(k_m * t_m))) * l_m);
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 4.1e-76)
		tmp = (l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m;
	else
		tmp = l_m * (l_m / ((((k_m * k_m) * t_m) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.1e-76], N[(N[(l$95$m / N[(N[(N[(k$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.0999999999999998e-76

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   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 57.7

         \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

      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. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      12. lower-*.f64 62.6

          \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. Applied rewrites 62.6%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      4. associate-*r* N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. lower-*.f64 66.0

         \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   10. Applied rewrites 66.0%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 4.0999999999999998e-76 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.9

         \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      7. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      14. associate-/l* N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

      16. lower-/.f64 55.2

          \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      17. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      18. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      19. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      20. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      21. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      22. cube-mult N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

      23. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

      24. associate-*r* N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      25. lower-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      26. lower-*.f64 57.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

   6. Applied rewrites 57.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot \color{blue}{t}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]

      5. lower-*.f64 61.2

         \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot t} \]

   8. Applied rewrites 61.2%

      \[\leadsto \ell \cdot \frac{\ell}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot t\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{l\_m}{\left(\left(k\_m \cdot t\_m\right) \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)} \cdot l\_m\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* (/ l_m (* (* (* k_m t_m) t_m) (* k_m t_m))) l_m)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m);
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m)
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m / Float64(Float64(Float64(k_m * t_m) * t_m) * Float64(k_m * t_m))) * l_m))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m / (((k_m * t_m) * t_m) * (k_m * t_m))) * l_m);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m / N[(N[(N[(k$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 50.9

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   7. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   14. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   16. lower-/.f64 55.2

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   17. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   18. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   19. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   20. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   21. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   22. cube-mult N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   23. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   24. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   25. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   26. lower-*.f64 57.7

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

6. Applied rewrites 57.7%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
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 57.7

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   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. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 62.6

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.6%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   6. lower-*.f64 66.0

      \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Applied rewrites 66.0%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 18: 65.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{l\_m}{t\_m \cdot \left(\left(k\_m \cdot t\_m\right) \cdot \left(k\_m \cdot t\_m\right)\right)} \cdot l\_m\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* (/ l_m (* t_m (* (* k_m t_m) (* k_m t_m)))) l_m)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m);
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m)
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m / Float64(t_m * Float64(Float64(k_m * t_m) * Float64(k_m * t_m)))) * l_m))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m / (t_m * ((k_m * t_m) * (k_m * t_m)))) * l_m);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m / N[(t$95$m * N[(N[(k$95$m * t$95$m), $MachinePrecision] * N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 50.9

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   7. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   14. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   16. lower-/.f64 55.2

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   17. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   18. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   19. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   20. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   21. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   22. cube-mult N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   23. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   24. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   25. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   26. lower-*.f64 57.7

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

6. Applied rewrites 57.7%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
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 57.7

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   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. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 62.6

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.6%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   4. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   7. associate-*l* N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   9. lower-*.f64 65.5

      \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

10. Applied rewrites 65.5%

    \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot \ell \]
11. Add Preprocessing

Alternative 19: 59.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{l\_m}{k\_m \cdot \left(k\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)} \cdot l\_m\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* (/ l_m (* k_m (* k_m (* (* t_m t_m) t_m)))) l_m)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m);
}

Fortran

l_m =     private
k_m =     private
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_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m)
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m);
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m)

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m / Float64(k_m * Float64(k_m * Float64(Float64(t_m * t_m) * t_m)))) * l_m))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m / (k_m * (k_m * ((t_m * t_m) * t_m)))) * l_m);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m / N[(k$95$m * N[(k$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 50.9

      \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   7. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   11. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   14. associate-/l* N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   15. lower-*.f64 N/A

       \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \]

   16. lower-/.f64 55.2

       \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   17. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   18. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

   19. pow2 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   20. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   21. lift-pow.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   22. cube-mult N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)} \]

   23. lift-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{t}\right)\right)} \]

   24. associate-*r* N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   25. lower-*.f64 N/A

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   26. lower-*.f64 57.7

       \[\leadsto \ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\color{blue}{t} \cdot t\right)} \]

6. Applied rewrites 57.7%

   \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
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 57.7

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\ell} \]

   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. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   12. lower-*.f64 62.6

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

8. Applied rewrites 62.6%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell \]

   6. pow2 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left({k}^{2} \cdot t\right)} \cdot \ell \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left({k}^{2} \cdot t\right)} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot {k}^{2}\right)} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{2}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot t\right) \cdot {k}^{2}} \cdot \ell \]

   11. unpow3 N/A

       \[\leadsto \frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell \]

   12. *-commutative N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{3}} \cdot \ell \]

   14. pow2 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}} \cdot \ell \]

   15. associate-*l* N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot {t}^{3}\right)} \cdot \ell \]

   18. unpow3 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

   20. lower-*.f64 59.5

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

10. Applied rewrites 59.5%

    \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\left(t \cdot t\right) \cdot t\right)\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025151 
(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))))