Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 83.6%

Time: 9.2s

Alternatives: 22

Speedup: 7.1×

• 26.7% 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: 54.1% 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: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;l\_m \leq 9.9093 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{{\sin k}^{2} \cdot t\_m}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= l_m 8.5e-162)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) k)
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= l_m 9.9093e+147)
      (/
       2.0
       (*
        (/
         (fma 2.0 (pow (* (sin k) t_m) 2.0) (pow (* (sin k) k) 2.0))
         (* (cos k) (* l_m l_m)))
        t_m))
      (/
       2.0
       (* (* (/ k l_m) (/ k l_m)) (/ (* (pow (sin k) 2.0) t_m) (cos k))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (l_m <= 8.5e-162) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (l_m <= 9.9093e+147) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t_m), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((pow(sin(k), 2.0) * t_m) / cos(k)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (l_m <= 8.5e-162)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (l_m <= 9.9093e+147)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t_m) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64((sin(k) ^ 2.0) * t_m) / cos(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[l$95$m, 8.5e-162], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 9.9093e+147], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 8.49999999999999955e-162

   1. Initial program 62.8%

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

      1. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.4

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \color{blue}{k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.4%

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

   if 8.49999999999999955e-162 < l < 9.90929999999999934e147

   1. Initial program 74.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.9%

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

   if 9.90929999999999934e147 < l

   1. Initial program 34.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 34.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 38.0

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

   11. Applied rewrites 38.0%

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

   12. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-pow.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-cos.f64 79.2

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

   14. Applied rewrites 79.2%

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

Alternative 2: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+208}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\_m\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{{\sin k}^{2} \cdot t\_m}{\cos k}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+208)
    (/
     2.0
     (*
      (/
       (fma 2.0 (pow (* (sin k) t_m) 2.0) (pow (* (sin k) k) 2.0))
       (* (cos k) (* l_m l_m)))
      t_m))
    (/
     2.0
     (* (* (/ k l_m) (/ k l_m)) (/ (* (pow (sin k) 2.0) t_m) (cos k)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k) * t_m), 2.0), pow((sin(k) * k), 2.0)) / (cos(k) * (l_m * l_m))) * t_m);
	} else {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((pow(sin(k), 2.0) * t_m) / cos(k)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k) * t_m) ^ 2.0), (Float64(sin(k) * k) ^ 2.0)) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64((sin(k) ^ 2.0) * t_m) / cos(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+208], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999999e208

   1. Initial program 85.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.9%

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

   if 3.9999999999999999e208 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   12. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-pow.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-cos.f64 80.2

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

   14. Applied rewrites 80.2%

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

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{{\sin k}^{2} \cdot t\_m}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+30)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* (pow (sin k) 2.0) t_m) (cos k))))
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k)) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.1e+30) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((pow(sin(k), 2.0) * t_m) / cos(k)));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.1e+30)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64((sin(k) ^ 2.0) * t_m) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e+30], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e30

   1. Initial program 58.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 40.8%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   12. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-pow.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-cos.f64 82.5

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

   14. Applied rewrites 82.5%

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

   if 2.1e30 < t

   1. Initial program 71.1%

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

      1. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 38.1

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

   4. Applied rewrites 38.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 38.1

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

   6. Applied rewrites 38.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{{\sin k}^{2} \cdot t\_m}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 1\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+30)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* (pow (sin k) 2.0) t_m) (cos k))))
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
      (+ (fma (/ k t_m) (/ k t_m) 1.0) 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.1e+30) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((pow(sin(k), 2.0) * t_m) / cos(k)));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * (fma((k / t_m), (k / t_m), 1.0) + 1.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.1e+30)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64((sin(k) ^ 2.0) * t_m) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 1.0) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e+30], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e30

   1. Initial program 58.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 40.8%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   12. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-pow.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-cos.f64 82.5

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

   14. Applied rewrites 82.5%

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

   if 2.1e30 < t

   1. Initial program 71.1%

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

      1. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 38.1

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

   4. Applied rewrites 38.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 38.1

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

   6. Applied rewrites 38.1%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right)} + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{{\sin k}^{2} \cdot t\_m}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log l\_m \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e+30)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* (pow (sin k) 2.0) t_m) (cos k))))
    (if (<= t_m 5.8e+102)
      (/
       2.0
       (*
        (* (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (/
       2.0
       (*
        (* (* (exp (- (* (log t_m) 3.0) (* (log l_m) 2.0))) (sin k)) (tan k))
        2.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.1e+30) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((pow(sin(k), 2.0) * t_m) / cos(k)));
	} else if (t_m <= 5.8e+102) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 2.1d+30) then
        tmp = 2.0d0 / (((k / l_m) * (k / l_m)) * (((sin(k) ** 2.0d0) * t_m) / cos(k)))
    else if (t_m <= 5.8d+102) then
        tmp = 2.0d0 / ((((((t_m ** 3.0d0) / l_m) / l_m) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l_m) * 2.0d0))) * sin(k)) * tan(k)) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 2.1e+30) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((Math.pow(Math.sin(k), 2.0) * t_m) / Math.cos(k)));
	} else if (t_m <= 5.8e+102) {
		tmp = 2.0 / (((((Math.pow(t_m, 3.0) / l_m) / l_m) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l_m) * 2.0))) * Math.sin(k)) * Math.tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 2.1e+30:
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((math.pow(math.sin(k), 2.0) * t_m) / math.cos(k)))
	elif t_m <= 5.8e+102:
		tmp = 2.0 / (((((math.pow(t_m, 3.0) / l_m) / l_m) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l_m) * 2.0))) * math.sin(k)) * math.tan(k)) * 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.1e+30)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64((sin(k) ^ 2.0) * t_m) / cos(k))));
	elseif (t_m <= 5.8e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 2.1e+30)
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * (((sin(k) ^ 2.0) * t_m) / cos(k)));
	elseif (t_m <= 5.8e+102)
		tmp = 2.0 / ((((((t_m ^ 3.0) / l_m) / l_m) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l_m) * 2.0))) * sin(k)) * tan(k)) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e+30], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+102], N[(2.0 / N[(N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.1e30

   1. Initial program 58.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 40.8%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   12. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-pow.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-cos.f64 82.5

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

   14. Applied rewrites 82.5%

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

   if 2.1e30 < t < 5.8000000000000005e102

   1. Initial program 82.1%

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

      1. lift-*.f64 N/A

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

      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-pow.f64 N/A

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

      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. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 95.5

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

   4. Applied rewrites 95.5%

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

   if 5.8000000000000005e102 < t

   1. Initial program 65.8%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 31.9

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

   7. Applied rewrites 31.9%

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

Alternative 6: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+208}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot k\right) \cdot k\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+208)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (* (* (/ k l_m) (/ k l_m)) k) k) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((((k / l_m) * (k / l_m)) * k) * k) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+208) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((((k / l_m) * (k / l_m)) * k) * k) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((((k / l_m) * (k / l_m)) * k) * k) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((((k / l_m) * (k / l_m)) * k) * k) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * k) * k) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((((k / l_m) * (k / l_m)) * k) * k) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+208], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999999e208

   1. Initial program 85.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 3.9999999999999999e208 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

   13. Applied rewrites 52.7%

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

Alternative 7: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+208}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+208)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+208) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+208], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999999e208

   1. Initial program 85.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 3.9999999999999999e208 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   13. Applied rewrites 52.7%

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

Alternative 8: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+208}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k \cdot k}{l\_m}}{l\_m} \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+208)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (/ (/ (* k k) l_m) l_m) (* k k)) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((((k * k) / l_m) / l_m) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+208) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((((k * k) / l_m) / l_m) * (k * k)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((((k * k) / l_m) / l_m) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((((k * k) / l_m) / l_m) * (k * k)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) / l_m) / l_m) * Float64(k * k)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((((k * k) / l_m) / l_m) * (k * k)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+208], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999999e208

   1. Initial program 85.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 3.9999999999999999e208 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 52.7

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

   13. Applied rewrites 52.7%

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

Alternative 9: 58.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+208}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{l\_m \cdot l\_m} \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+208)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (/ (* k k) (* l_m l_m)) (* k k)) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((((k * k) / (l_m * l_m)) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+208) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / ((((k * k) / (l_m * l_m)) * (k * k)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((((k * k) / (l_m * l_m)) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / ((((k * k) / (l_m * l_m)) * (k * k)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / Float64(l_m * l_m)) * Float64(k * k)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / ((((k * k) / (l_m * l_m)) * (k * k)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+208], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999999e208

   1. Initial program 85.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 3.9999999999999999e208 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

Alternative 10: 58.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+208}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot \frac{k}{l\_m \cdot l\_m}\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l_m l_m)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+208)
    (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))
    (/ 2.0 (* (* (* k (/ k (* l_m l_m))) (* k k)) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * (k / (l_m * l_m))) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+208) then
        tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((k * (k / (l_m * l_m))) * (k * k)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208) {
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k * (k / (l_m * l_m))) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+208:
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((k * (k / (l_m * l_m))) * (k * k)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / Float64(l_m * l_m))) * Float64(k * k)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+208)
		tmp = (l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((k * (k / (l_m * l_m))) * (k * k)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+208], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 3.9999999999999999e208

   1. Initial program 85.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 76.5

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

   5. Applied rewrites 76.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   if 3.9999999999999999e208 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 27.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 46.8

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

   13. Applied rewrites 46.8%

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

Alternative 11: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{+31}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right) \cdot \frac{{\sin k}^{2} \cdot t\_m}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e+31)
    (/ 2.0 (* (* (/ k l_m) (/ k l_m)) (/ (* (pow (sin k) 2.0) t_m) (cos k))))
    (/
     2.0
     (*
      (/
       (*
        (pow
         (*
          (*
           (fma
            (- (* 0.008333333333333333 (* k k)) 0.16666666666666666)
            (* k k)
            1.0)
           k)
          t_m)
         2.0)
        2.0)
       (* (cos k) (* l_m l_m)))
      t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1e+31) {
		tmp = 2.0 / (((k / l_m) * (k / l_m)) * ((pow(sin(k), 2.0) * t_m) / cos(k)));
	} else {
		tmp = 2.0 / (((pow(((fma(((0.008333333333333333 * (k * k)) - 0.16666666666666666), (k * k), 1.0) * k) * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1e+31)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l_m) * Float64(k / l_m)) * Float64(Float64((sin(k) ^ 2.0) * t_m) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(k * k)) - 0.16666666666666666), Float64(k * k), 1.0) * k) * t_m) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1e+31], N[(2.0 / N[(N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[(N[(N[(N[(0.008333333333333333 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999996e30

   1. Initial program 58.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 40.8%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 62.7

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

   11. Applied rewrites 62.7%

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

   12. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-pow.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-cos.f64 82.5

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

   14. Applied rewrites 82.5%

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

   if 9.9999999999999996e30 < t

   1. Initial program 71.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 72.9

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{\left(\left(k \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right)\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left({k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) \cdot {k}^{2} + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       11. lift-*.f64 75.7

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 75.7%

       \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{l\_m \cdot l\_m}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t\_m}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e+25)
    (* (* (/ (* l_m l_m) (* k k)) (/ (cos k) (* (pow (sin k) 2.0) t_m))) 2.0)
    (/
     2.0
     (*
      (/
       (*
        (pow
         (*
          (*
           (fma
            (- (* 0.008333333333333333 (* k k)) 0.16666666666666666)
            (* k k)
            1.0)
           k)
          t_m)
         2.0)
        2.0)
       (* (cos k) (* l_m l_m)))
      t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.9e+25) {
		tmp = (((l_m * l_m) / (k * k)) * (cos(k) / (pow(sin(k), 2.0) * t_m))) * 2.0;
	} else {
		tmp = 2.0 / (((pow(((fma(((0.008333333333333333 * (k * k)) - 0.16666666666666666), (k * k), 1.0) * k) * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.9e+25)
		tmp = Float64(Float64(Float64(Float64(l_m * l_m) / Float64(k * k)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t_m))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(k * k)) - 0.16666666666666666), Float64(k * k), 1.0) * k) * t_m) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e+25], N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[(N[(N[(N[(0.008333333333333333 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999999e25

   1. Initial program 58.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.6%

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

   if 2.8999999999999999e25 < t

   1. Initial program 71.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 73.3

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

   8. Applied rewrites 73.3%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{\left(\left(k \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right)\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left({k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) \cdot {k}^{2} + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       11. lift-*.f64 76.1

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 76.1%

       \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k \cdot \left(l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{t\_2} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\_m\right)}^{2} \cdot 2}{t\_2} \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (cos k) (* l_m l_m))))
   (*
    t_s
    (if (<= t_m 2.9e+25)
      (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) t_2) t_m))
      (/
       2.0
       (*
        (/
         (*
          (pow
           (*
            (*
             (fma
              (- (* 0.008333333333333333 (* k k)) 0.16666666666666666)
              (* k k)
              1.0)
             k)
            t_m)
           2.0)
          2.0)
         t_2)
        t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = cos(k) * (l_m * l_m);
	double tmp;
	if (t_m <= 2.9e+25) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / t_2) * t_m);
	} else {
		tmp = 2.0 / (((pow(((fma(((0.008333333333333333 * (k * k)) - 0.16666666666666666), (k * k), 1.0) * k) * t_m), 2.0) * 2.0) / t_2) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(cos(k) * Float64(l_m * l_m))
	tmp = 0.0
	if (t_m <= 2.9e+25)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / t_2) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(k * k)) - 0.16666666666666666), Float64(k * k), 1.0) * k) * t_m) ^ 2.0) * 2.0) / t_2) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e+25], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[(N[(N[(N[(0.008333333333333333 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999999e25

   1. Initial program 58.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 68.5

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

   8. Applied rewrites 68.5%

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

   if 2.8999999999999999e25 < t

   1. Initial program 71.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 73.3

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

   8. Applied rewrites 73.3%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{\left(\left(k \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right)\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left({k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) \cdot {k}^{2} + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       11. lift-*.f64 76.1

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 76.1%

       \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\left(\cos k \cdot l\_m\right) \cdot l\_m} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e+25)
    (/ 2.0 (* (/ (pow (* (sin k) k) 2.0) (* (* (cos k) l_m) l_m)) t_m))
    (/
     2.0
     (*
      (/
       (*
        (pow
         (*
          (*
           (fma
            (- (* 0.008333333333333333 (* k k)) 0.16666666666666666)
            (* k k)
            1.0)
           k)
          t_m)
         2.0)
        2.0)
       (* (cos k) (* l_m l_m)))
      t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.9e+25) {
		tmp = 2.0 / ((pow((sin(k) * k), 2.0) / ((cos(k) * l_m) * l_m)) * t_m);
	} else {
		tmp = 2.0 / (((pow(((fma(((0.008333333333333333 * (k * k)) - 0.16666666666666666), (k * k), 1.0) * k) * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.9e+25)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k) * k) ^ 2.0) / Float64(Float64(cos(k) * l_m) * l_m)) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(k * k)) - 0.16666666666666666), Float64(k * k), 1.0) * k) * t_m) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e+25], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[(N[(N[(N[(0.008333333333333333 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999999e25

   1. Initial program 58.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. metadata-eval N/A

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

      6. pow-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 77.6

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

   7. Applied rewrites 77.6%

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

   8. Taylor expanded in t around 0

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

      1. pow-flip N/A

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

      2. metadata-eval N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   10. Applied rewrites 68.5%

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

   if 2.8999999999999999e25 < t

   1. Initial program 71.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 73.3

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

   8. Applied rewrites 73.3%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{\left(\left(k \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right)\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left({k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) \cdot {k}^{2} + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       11. lift-*.f64 76.1

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 76.1%

       \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 70.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m \cdot l\_m} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + \frac{k \cdot k}{t\_m \cdot t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.6e-97)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (if (<= t_m 5e+16)
      (/
       2.0
       (*
        (* (* (/ (* (* t_m t_m) t_m) (* l_m l_m)) (sin k)) (tan k))
        (+ (+ 1.0 (/ (* k k) (* t_m t_m))) 1.0)))
      (/
       2.0
       (*
        (/
         (*
          (pow
           (*
            (*
             (fma
              (- (* 0.008333333333333333 (* k k)) 0.16666666666666666)
              (* k k)
              1.0)
             k)
            t_m)
           2.0)
          2.0)
         (* (cos k) (* l_m l_m)))
        t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 4.6e-97) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else if (t_m <= 5e+16) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / (l_m * l_m)) * sin(k)) * tan(k)) * ((1.0 + ((k * k) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((pow(((fma(((0.008333333333333333 * (k * k)) - 0.16666666666666666), (k * k), 1.0) * k) * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 4.6e-97)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	elseif (t_m <= 5e+16)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / Float64(l_m * l_m)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + Float64(Float64(k * k) / Float64(t_m * t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(k * k)) - 0.16666666666666666), Float64(k * k), 1.0) * k) * t_m) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.6e-97], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+16], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[(N[(N[(N[(0.008333333333333333 * N[(k * k), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.59999999999999988e-97

   1. Initial program 55.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 40.7%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 64.2

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

   11. Applied rewrites 64.2%

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

   13. Applied rewrites 65.7%

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

   if 4.59999999999999988e-97 < t < 5e16

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 73.4

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

   6. Applied rewrites 73.4%

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

   if 5e16 < t

   1. Initial program 72.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{\left(\left(k \cdot \left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right)\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(1 + {k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right)\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       3. +-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left({k}^{2} \cdot \left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\left(\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}\right) \cdot {k}^{2} + 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {k}^{2} - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, {k}^{2}, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       10. pow2 N/A

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(k \cdot k\right) - \frac{1}{6}, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

       11. lift-*.f64 76.7

           \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   11. Applied rewrites 76.7%

       \[\leadsto \frac{2}{\frac{{\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(k \cdot k\right) - 0.16666666666666666, k \cdot k, 1\right) \cdot k\right) \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 68.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\cos k \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.6e-98)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (/ 2.0 (* (/ (* (pow (* k t_m) 2.0) 2.0) (* (cos k) (* l_m l_m))) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 3.6e-98) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 3.6d-98) then
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) / (cos(k) * (l_m * l_m))) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 3.6e-98) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) / (Math.cos(k) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 3.6e-98:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) / (math.cos(k) * (l_m * l_m))) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 3.6e-98)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / Float64(cos(k) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 3.6e-98)
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) / (cos(k) * (l_m * l_m))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-98], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.6000000000000002e-98

   1. Initial program 55.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 40.7%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 64.2

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

   11. Applied rewrites 64.2%

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

   13. Applied rewrites 65.7%

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

   if 3.6000000000000002e-98 < t

   1. Initial program 72.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 17: 68.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{l\_m \cdot l\_m} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.02e-81)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) (* l_m l_m)) 2.0) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.02e-81) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 1.02d-81) then
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / (l_m * l_m)) * 2.0d0) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 1.02e-81) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 1.02e-81:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / (l_m * l_m)) * 2.0) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.02e-81)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / Float64(l_m * l_m)) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.02e-81)
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / (l_m * l_m)) * 2.0) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.02e-81], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.01999999999999998e-81

   1. Initial program 56.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 41.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 64.7

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

   11. Applied rewrites 64.7%

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

   13. Applied rewrites 66.1%

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

   if 1.01999999999999998e-81 < t

   1. Initial program 71.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

Alternative 18: 67.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e-79)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (/ (* l_m l_m) (* (pow (* k t_m) 2.0) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.5e-79) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = (l_m * l_m) / (pow((k * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 7.5d-79) then
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = (l_m * l_m) / (((k * t_m) ** 2.0d0) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 7.5e-79) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = (l_m * l_m) / (Math.pow((k * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.5e-79:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	else:
		tmp = (l_m * l_m) / (math.pow((k * t_m), 2.0) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.5e-79)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.5e-79)
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = (l_m * l_m) / (((k * t_m) ^ 2.0) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-79], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999969e-79

   1. Initial program 56.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 41.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 64.7

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

   11. Applied rewrites 64.7%

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

   13. Applied rewrites 66.1%

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

   if 7.49999999999999969e-79 < t

   1. Initial program 71.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 59.8

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

   7. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

Alternative 19: 63.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{k \cdot \left(k \cdot {t\_m}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-79)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (/ (* l_m l_m) (* k (* k (pow t_m 3.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.8e-79) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = (l_m * l_m) / (k * (k * pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 7.8d-79) then
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = (l_m * l_m) / (k * (k * (t_m ** 3.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 7.8e-79) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = (l_m * l_m) / (k * (k * Math.pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.8e-79:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	else:
		tmp = (l_m * l_m) / (k * (k * math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.8e-79)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64(k * Float64(k * (t_m ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.8e-79)
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = (l_m * l_m) / (k * (k * (t_m ^ 3.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-79], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.80000000000000011e-79

   1. Initial program 56.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 41.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 64.7

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

   11. Applied rewrites 64.7%

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

   13. Applied rewrites 66.1%

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

   if 7.80000000000000011e-79 < t

   1. Initial program 71.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 64.5

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

   7. Applied rewrites 64.5%

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

Alternative 20: 62.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{l\_m}{\left(k \cdot k\right) \cdot {t\_m}^{3}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.6e-55)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (* l_m (/ l_m (* (* k k) (pow t_m 3.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 5.6e-55) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = l_m * (l_m / ((k * k) * pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 5.6d-55) then
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = l_m * (l_m / ((k * k) * (t_m ** 3.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 5.6e-55) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = l_m * (l_m / ((k * k) * Math.pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 5.6e-55:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	else:
		tmp = l_m * (l_m / ((k * k) * math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 5.6e-55)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	else
		tmp = Float64(l_m * Float64(l_m / Float64(Float64(k * k) * (t_m ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 5.6e-55)
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = l_m * (l_m / ((k * k) * (t_m ^ 3.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 5.6e-55], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(l$95$m * N[(l$95$m / N[(N[(k * k), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.59999999999999968e-55

   1. Initial program 57.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 41.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 64.5

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

   11. Applied rewrites 64.5%

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

   13. Applied rewrites 65.8%

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

   if 5.59999999999999968e-55 < t

   1. Initial program 69.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 59.8

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

   5. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 66.7

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 63.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{l\_m} \cdot \frac{k}{l\_m}\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{2}{l\_m}\right) \cdot \left(k \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.45e-56)
    (/ 2.0 (* (* (* k k) (* (/ k l_m) (/ k l_m))) t_m))
    (/ 2.0 (* (* (* (/ (* t_m t_m) l_m) (/ 2.0 l_m)) (* k k)) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.45e-56) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((((t_m * t_m) / l_m) * (2.0 / l_m)) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Fortran

l_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)
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
    real(8) :: tmp
    if (t_m <= 1.45d-56) then
        tmp = 2.0d0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
    else
        tmp = 2.0d0 / (((((t_m * t_m) / l_m) * (2.0d0 / l_m)) * (k * k)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (t_m <= 1.45e-56) {
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	} else {
		tmp = 2.0 / (((((t_m * t_m) / l_m) * (2.0 / l_m)) * (k * k)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 1.45e-56:
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m)
	else:
		tmp = 2.0 / (((((t_m * t_m) / l_m) * (2.0 / l_m)) * (k * k)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.45e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k / l_m) * Float64(k / l_m))) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(2.0 / l_m)) * Float64(k * k)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.45e-56)
		tmp = 2.0 / (((k * k) * ((k / l_m) * (k / l_m))) * t_m);
	else
		tmp = 2.0 / (((((t_m * t_m) / l_m) * (2.0 / l_m)) * (k * k)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-56], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k / l$95$m), $MachinePrecision] * N[(k / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(2.0 / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999996e-56

   1. Initial program 57.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 79.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. lift-*.f64 64.5

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. Applied rewrites 64.5%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.8%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \color{blue}{t}} \]

   if 1.44999999999999996e-56 < t

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 74.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

   8. Applied rewrites 22.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. lift-*.f64 50.7

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. Applied rewrites 50.7%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{2}{\left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{{t}^{2} \cdot 2}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       7. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       9. lower-/.f64 63.5

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   14. Applied rewrites 63.5%

       \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 50.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{l\_m \cdot l\_m}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ (* l_m l_m) (* (* k k) (* (* t_m t_m) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)));
}

Fortran

l_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)
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
    code = t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)))
end function

Java

l_m = Math.abs(l);
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) {
	return t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(l_m * l_m) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * ((l_m * l_m) / ((k * k) * ((t_m * t_m) * t_m)));
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 58.3

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 58.3%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. unpow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. lift-*.f64 58.3

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

7. Applied rewrites 58.3%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025061 
(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))))