Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 81.6%

Time: 8.9s

Alternatives: 20

Speedup: 8.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k\_m \cdot t\_m\right)}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.2e-129)
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m)) (tan k_m))
      2.0))
    (/
     2.0
     (*
      (/
       (fma (pow (* (sin k_m) t_m) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
       (* (cos k_m) l_m))
      (/ t_m l_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.2e-129) {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / ((fma(pow((sin(k_m) * t_m), 2.0), 2.0, pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * l_m)) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.2e-129)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k_m) * t_m) ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.2e-129], 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$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999994e-129

   1. Initial program 49.9%

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

   3. Taylor expanded in t around 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 47.4%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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 15.9

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

   7. Applied rewrites 15.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 15.9

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

   9. Applied rewrites 15.9%

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

   if 1.19999999999999994e-129 < k

   1. Initial program 51.4%

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 69.5%

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

   9. Applied rewrites 79.7%

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

Alternative 2: 66.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t_m), 2.0)) + 1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = (l_m * l_m) / (k_m * (k_m * Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * (k_m * k_m)) * k_m) * k_m);
	}
	return t_s * tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.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))) < +inf.0

   1. Initial program 78.7%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 70.9%

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

   if +inf.0 < (*.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 0.0%

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 14.8%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 33.9%

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

   13. Applied rewrites 38.4%

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

Alternative 3: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log l\_m\right)} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-60)
    (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) (* (cos k_m) l_m)) (/ t_m l_m)))
    (if (<= t_m 9.2e+89)
      (/
       2.0
       (*
        (* (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (/
       2.0
       (*
        (*
         (* (exp (fma (log t_m) 3.0 (* -2.0 (log l_m)))) (sin k_m))
         (tan k_m))
        2.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 6.2e-60) {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 9.2e+89) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * 2.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 6.2e-60)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	elseif (t_m <= 9.2e+89)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l_m)))) * sin(k_m)) * tan(k_m)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-60], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+89], 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$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999976e-60

   1. Initial program 41.7%

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 80.7%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 67.8%

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

   if 6.19999999999999976e-60 < t < 9.1999999999999996e89

   1. Initial program 70.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 83.3

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

   4. Applied rewrites 83.3%

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

   if 9.1999999999999996e89 < t

   1. Initial program 73.1%

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

   3. Taylor expanded in t around 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 73.1%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   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 52.0

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

   7. Applied rewrites 52.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 52.1

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

   9. Applied rewrites 52.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0))) <= 1e+30) {
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (t_m / (l_m * l_m))) * (k_m * k_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+30) then
        tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / (((k_m * k_m) * (t_m / (l_m * l_m))) * (k_m * k_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t_m), 2.0)) + 1.0))) <= 1e+30) {
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / (((k_m * k_m) * (t_m / (l_m * l_m))) * (k_m * k_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t_m), 2.0)) + 1.0))) <= 1e+30:
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / (((k_m * k_m) * (t_m / (l_m * l_m))) * (k_m * k_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0))) <= 1e+30)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(t_m / Float64(l_m * l_m))) * Float64(k_m * k_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t_m) ^ 2.0)) + 1.0))) <= 1e+30)
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / (((k_m * k_m) * (t_m / (l_m * l_m))) * (k_m * k_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[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$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+30], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $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)))) < 1e30

   1. Initial program 78.8%

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 63.0%

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

   if 1e30 < (/.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 19.8%

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 29.2%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 41.8%

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

Alternative 5: 57.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0))) <= 1e+30) {
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t_m) * (k_m * k_m)) / (l_m * l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+30) then
        tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m))
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t_m) * (k_m * k_m)) / (l_m * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l_m * l_m)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t_m), 2.0)) + 1.0))) <= 1e+30) {
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m));
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t_m) * (k_m * k_m)) / (l_m * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l_m * l_m)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t_m), 2.0)) + 1.0))) <= 1e+30:
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m))
	else:
		tmp = 2.0 / ((((k_m * k_m) * t_m) * (k_m * k_m)) / (l_m * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l_m * l_m)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0))) <= 1e+30)
		tmp = Float64(Float64(l_m * l_m) / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * t_m) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t_m) * Float64(k_m * k_m)) / Float64(l_m * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l_m * l_m)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t_m) ^ 2.0)) + 1.0))) <= 1e+30)
		tmp = (l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m));
	else
		tmp = 2.0 / ((((k_m * k_m) * t_m) * (k_m * k_m)) / (l_m * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[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$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+30], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $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)))) < 1e30

   1. Initial program 78.8%

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 63.0%

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

   if 1e30 < (/.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 19.8%

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 29.2%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in l around 0

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

   11. Applied rewrites 36.1%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), t, {t}^{3}\right), k \cdot k, {t}^{3} \cdot 2\right) \cdot \left(k \cdot k\right)}{\ell \cdot \color{blue}{\ell}}} \]

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 40.2%

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

Alternative 6: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 10^{+88}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-60)
    (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) (* (cos k_m) l_m)) (/ t_m l_m)))
    (if (<= t_m 1e+88)
      (/
       2.0
       (*
        (* (* (/ (/ (pow t_m 3.0) l_m) l_m) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t_m) 2.0)) 1.0)))
      (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 6.2e-60) {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 1e+88) {
		tmp = 2.0 / (((((pow(t_m, 3.0) / l_m) / l_m) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 6.2d-60) then
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) / (cos(k_m) * l_m)) * (t_m / l_m))
    else if (t_m <= 1d+88) then
        tmp = 2.0d0 / ((((((t_m ** 3.0d0) / l_m) / l_m) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 6.2e-60) {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) / (Math.cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 1e+88) {
		tmp = 2.0 / (((((Math.pow(t_m, 3.0) / l_m) / l_m) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 6.2e-60:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) / (math.cos(k_m) * l_m)) * (t_m / l_m))
	elif t_m <= 1e+88:
		tmp = 2.0 / (((((math.pow(t_m, 3.0) / l_m) / l_m) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t_m), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 6.2e-60)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	elseif (t_m <= 1e+88)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((t_m ^ 3.0) / l_m) / l_m) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 6.2e-60)
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	elseif (t_m <= 1e+88)
		tmp = 2.0 / ((((((t_m ^ 3.0) / l_m) / l_m) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t_m) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-60], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+88], 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$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999976e-60

   1. Initial program 41.7%

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 80.7%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 67.8%

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

   if 6.19999999999999976e-60 < t < 9.99999999999999959e87

   1. Initial program 70.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 83.3

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

   4. Applied rewrites 83.3%

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

   if 9.99999999999999959e87 < t

   1. Initial program 73.1%

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 91.2%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 91.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+88}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 10^{+88}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \sin k\_m\right) \cdot \left(\tan k\_m \cdot \left(\left({\left(\frac{k\_m}{t\_m}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-60)
    (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) (* (cos k_m) l_m)) (/ t_m l_m)))
    (if (<= t_m 1e+88)
      (/
       2.0
       (*
        (* (/ (/ (* (* t_m t_m) t_m) l_m) l_m) (sin k_m))
        (* (tan k_m) (+ (+ (pow (/ k_m t_m) 2.0) 1.0) 1.0))))
      (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5.5e-60) {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 1e+88) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l_m) / l_m) * sin(k_m)) * (tan(k_m) * ((pow((k_m / t_m), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5.5d-60) then
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) / (cos(k_m) * l_m)) * (t_m / l_m))
    else if (t_m <= 1d+88) then
        tmp = 2.0d0 / ((((((t_m * t_m) * t_m) / l_m) / l_m) * sin(k_m)) * (tan(k_m) * ((((k_m / t_m) ** 2.0d0) + 1.0d0) + 1.0d0)))
    else
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5.5e-60) {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) / (Math.cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 1e+88) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l_m) / l_m) * Math.sin(k_m)) * (Math.tan(k_m) * ((Math.pow((k_m / t_m), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 5.5e-60:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) / (math.cos(k_m) * l_m)) * (t_m / l_m))
	elif t_m <= 1e+88:
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l_m) / l_m) * math.sin(k_m)) * (math.tan(k_m) * ((math.pow((k_m / t_m), 2.0) + 1.0) + 1.0)))
	else:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5.5e-60)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	elseif (t_m <= 1e+88)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * sin(k_m)) * Float64(tan(k_m) * Float64(Float64((Float64(k_m / t_m) ^ 2.0) + 1.0) + 1.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 5.5e-60)
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	elseif (t_m <= 1e+88)
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l_m) / l_m) * sin(k_m)) * (tan(k_m) * ((((k_m / t_m) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.5e-60], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+88], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.4999999999999997e-60

   1. Initial program 41.7%

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 80.7%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 67.8%

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

   if 5.4999999999999997e-60 < t < 9.99999999999999959e87

   1. Initial program 70.2%

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

   4. Applied rewrites 83.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   if 9.99999999999999959e87 < t

   1. Initial program 73.1%

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 91.2%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 91.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+88}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{l\_m}}{\cos k\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k\_m \cdot {t\_m}^{3}}{l\_m}}{l\_m} \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-33)
    (/ 2.0 (* (/ (/ (pow (* (sin k_m) k_m) 2.0) l_m) (cos k_m)) (/ t_m l_m)))
    (if (<= t_m 2.7e+81)
      (/ 2.0 (* (* (/ (/ (* k_m (pow t_m 3.0)) l_m) l_m) (tan k_m)) 2.0))
      (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.4e-33) {
		tmp = 2.0 / (((pow((sin(k_m) * k_m), 2.0) / l_m) / cos(k_m)) * (t_m / l_m));
	} else if (t_m <= 2.7e+81) {
		tmp = 2.0 / (((((k_m * pow(t_m, 3.0)) / l_m) / l_m) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.4d-33) then
        tmp = 2.0d0 / (((((sin(k_m) * k_m) ** 2.0d0) / l_m) / cos(k_m)) * (t_m / l_m))
    else if (t_m <= 2.7d+81) then
        tmp = 2.0d0 / (((((k_m * (t_m ** 3.0d0)) / l_m) / l_m) * tan(k_m)) * 2.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.4e-33) {
		tmp = 2.0 / (((Math.pow((Math.sin(k_m) * k_m), 2.0) / l_m) / Math.cos(k_m)) * (t_m / l_m));
	} else if (t_m <= 2.7e+81) {
		tmp = 2.0 / (((((k_m * Math.pow(t_m, 3.0)) / l_m) / l_m) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 2.4e-33:
		tmp = 2.0 / (((math.pow((math.sin(k_m) * k_m), 2.0) / l_m) / math.cos(k_m)) * (t_m / l_m))
	elif t_m <= 2.7e+81:
		tmp = 2.0 / (((((k_m * math.pow(t_m, 3.0)) / l_m) / l_m) * math.tan(k_m)) * 2.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.4e-33)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / l_m) / cos(k_m)) * Float64(t_m / l_m)));
	elseif (t_m <= 2.7e+81)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * (t_m ^ 3.0)) / l_m) / l_m) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 2.4e-33)
		tmp = 2.0 / (((((sin(k_m) * k_m) ^ 2.0) / l_m) / cos(k_m)) * (t_m / l_m));
	elseif (t_m <= 2.7e+81)
		tmp = 2.0 / (((((k_m * (t_m ^ 3.0)) / l_m) / l_m) * tan(k_m)) * 2.0);
	else
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-33], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e+81], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.4e-33

   1. Initial program 43.1%

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 71.5%

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 68.8%

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

   if 2.4e-33 < t < 2.6999999999999999e81

   1. Initial program 72.1%

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

   3. Taylor expanded in t around 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 51.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 58.7%

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

   10. Applied rewrites 77.0%

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

   if 2.6999999999999999e81 < t

   1. Initial program 70.2%

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 82.9%

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

   9. Applied rewrites 89.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 89.2%

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

Alternative 9: 80.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k\_m \cdot {t\_m}^{3}}{l\_m}}{l\_m} \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-33)
    (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) (* (cos k_m) l_m)) (/ t_m l_m)))
    (if (<= t_m 2.7e+81)
      (/ 2.0 (* (* (/ (/ (* k_m (pow t_m 3.0)) l_m) l_m) (tan k_m)) 2.0))
      (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.4e-33) {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 2.7e+81) {
		tmp = 2.0 / (((((k_m * pow(t_m, 3.0)) / l_m) / l_m) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.4d-33) then
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) / (cos(k_m) * l_m)) * (t_m / l_m))
    else if (t_m <= 2.7d+81) then
        tmp = 2.0d0 / (((((k_m * (t_m ** 3.0d0)) / l_m) / l_m) * tan(k_m)) * 2.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.4e-33) {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) / (Math.cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 2.7e+81) {
		tmp = 2.0 / (((((k_m * Math.pow(t_m, 3.0)) / l_m) / l_m) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 2.4e-33:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) / (math.cos(k_m) * l_m)) * (t_m / l_m))
	elif t_m <= 2.7e+81:
		tmp = 2.0 / (((((k_m * math.pow(t_m, 3.0)) / l_m) / l_m) * math.tan(k_m)) * 2.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.4e-33)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	elseif (t_m <= 2.7e+81)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * (t_m ^ 3.0)) / l_m) / l_m) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 2.4e-33)
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	elseif (t_m <= 2.7e+81)
		tmp = 2.0 / (((((k_m * (t_m ^ 3.0)) / l_m) / l_m) * tan(k_m)) * 2.0);
	else
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-33], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e+81], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.4e-33

   1. Initial program 43.1%

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 71.5%

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 68.8%

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

   if 2.4e-33 < t < 2.6999999999999999e81

   1. Initial program 72.1%

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

   3. Taylor expanded in t around 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 51.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 58.7%

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

   10. Applied rewrites 77.0%

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

   if 2.6999999999999999e81 < t

   1. Initial program 70.2%

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 82.9%

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

   9. Applied rewrites 89.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 89.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k \cdot {t}^{3}}{\ell}}{\ell} \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.175:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\_m\right)}^{2} \cdot 2}{t\_2} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2} \cdot t\_m}{t\_2 \cdot l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (cos k_m) l_m)))
   (*
    t_s
    (if (<= k_m 0.175)
      (/ 2.0 (* (/ (* (pow (* k_m t_m) 2.0) 2.0) t_2) (/ t_m l_m)))
      (/ 2.0 (/ (* (pow (* (sin k_m) k_m) 2.0) t_m) (* t_2 l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cos(k_m) * l_m;
	double tmp;
	if (k_m <= 0.175) {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) * t_m) / (t_2 * l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) * l_m
    if (k_m <= 0.175d0) then
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) * 2.0d0) / t_2) * (t_m / l_m))
    else
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) * t_m) / (t_2 * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = Math.cos(k_m) * l_m;
	double tmp;
	if (k_m <= 0.175) {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) * t_m) / (t_2 * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	t_2 = math.cos(k_m) * l_m
	tmp = 0
	if k_m <= 0.175:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m))
	else:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) * t_m) / (t_2 * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(cos(k_m) * l_m)
	tmp = 0.0
	if (k_m <= 0.175)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) * 2.0) / t_2) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) * t_m) / Float64(t_2 * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	t_2 = cos(k_m) * l_m;
	tmp = 0.0;
	if (k_m <= 0.175)
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) * 2.0) / t_2) * (t_m / l_m));
	else
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) * t_m) / (t_2 * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.175], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(t$95$2 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.17499999999999999

   1. Initial program 52.6%

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.8%

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

   9. Applied rewrites 85.9%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 66.3%

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

   if 0.17499999999999999 < k

   1. Initial program 44.5%

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 66.5%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.175:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2} \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\_m\right)}^{2} \cdot 2}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m \cdot \left(l\_m \cdot l\_m\right)} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.02e-16)
    (/ 2.0 (* (/ (* (pow (* k_m t_m) 2.0) 2.0) (* (cos k_m) l_m)) (/ t_m l_m)))
    (/
     2.0
     (* (/ (pow (* (sin k_m) k_m) 2.0) (* (cos k_m) (* l_m l_m))) t_m)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.02e-16) {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) * 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / (cos(k_m) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.02d-16) then
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) * 2.0d0) / (cos(k_m) * l_m)) * (t_m / l_m))
    else
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) / (cos(k_m) * (l_m * l_m))) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.02e-16) {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) * 2.0) / (Math.cos(k_m) * l_m)) * (t_m / l_m));
	} else {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) / (Math.cos(k_m) * (l_m * l_m))) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 1.02e-16:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) * 2.0) / (math.cos(k_m) * l_m)) * (t_m / l_m))
	else:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) / (math.cos(k_m) * (l_m * l_m))) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.02e-16)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) * 2.0) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / Float64(cos(k_m) * Float64(l_m * l_m))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 1.02e-16)
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) * 2.0) / (cos(k_m) * l_m)) * (t_m / l_m));
	else
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) / (cos(k_m) * (l_m * l_m))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.02e-16], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Cos[k$95$m], $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 k < 1.0200000000000001e-16

   1. Initial program 53.2%

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 75.6%

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

   9. Applied rewrites 85.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 66.0%

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

   if 1.0200000000000001e-16 < k

   1. Initial program 43.4%

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in t around 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

   8. Applied rewrites 63.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k\_m \cdot l\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.22 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k\_m \cdot k\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{t\_2} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t\_m \cdot t\_m\right) \cdot t\_m}{l\_m}}{l\_m} \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right) \cdot \left(\left(1 + \frac{k\_m \cdot k\_m}{t\_m \cdot t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\_m\right)}^{2} \cdot 2}{t\_2} \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (cos k_m) l_m)))
   (*
    t_s
    (if (<= t_m 1.22e-101)
      (/
       2.0
       (*
        (/
         (*
          (fma
           (fma (* t_m t_m) -0.6666666666666666 1.0)
           (* k_m k_m)
           (* (* t_m t_m) 2.0))
          (* k_m k_m))
         t_2)
        (/ t_m l_m)))
      (if (<= t_m 9.2e+64)
        (/
         2.0
         (*
          (* (/ (/ (* (* t_m t_m) t_m) l_m) l_m) (* (sin k_m) (tan k_m)))
          (+ (+ 1.0 (/ (* k_m k_m) (* t_m t_m))) 1.0)))
        (/ 2.0 (* (/ (* (pow (* k_m t_m) 2.0) 2.0) t_2) (/ t_m l_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cos(k_m) * l_m;
	double tmp;
	if (t_m <= 1.22e-101) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k_m * k_m), ((t_m * t_m) * 2.0)) * (k_m * k_m)) / t_2) * (t_m / l_m));
	} else if (t_m <= 9.2e+64) {
		tmp = 2.0 / ((((((t_m * t_m) * t_m) / l_m) / l_m) * (sin(k_m) * tan(k_m))) * ((1.0 + ((k_m * k_m) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) * 2.0) / t_2) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(cos(k_m) * l_m)
	tmp = 0.0
	if (t_m <= 1.22e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k_m * k_m), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k_m * k_m)) / t_2) * Float64(t_m / l_m)));
	elseif (t_m <= 9.2e+64)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * t_m) / l_m) / l_m) * Float64(sin(k_m) * tan(k_m))) * Float64(Float64(1.0 + Float64(Float64(k_m * k_m) / Float64(t_m * t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) * 2.0) / t_2) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.22e-101], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+64], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k$95$m * k$95$m), $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[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.2199999999999999e-101

   1. Initial program 41.9%

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 82.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 50.3%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}} \]

   if 1.2199999999999999e-101 < t < 9.2e64

   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. 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 65.9

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-tan.f64 73.7

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

   6. Applied rewrites 73.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if 9.2e64 < t

   1. Initial program 68.8%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 81.2%

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

   9. Applied rewrites 87.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 87.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{\left(t \cdot t\right) \cdot t}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + \frac{k \cdot k}{t \cdot t}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k\_m \cdot k\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{l\_m} \cdot \frac{{t\_m}^{3}}{l\_m}\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.6e-105)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma (* t_m t_m) -0.6666666666666666 1.0)
         (* k_m k_m)
         (* (* t_m t_m) 2.0))
        (* k_m k_m))
       (* (cos k_m) l_m))
      (/ t_m l_m)))
    (if (<= t_m 6.5e+86)
      (/ 2.0 (* (* (* (/ k_m l_m) (/ (pow t_m 3.0) l_m)) (tan k_m)) 2.0))
      (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.6e-105) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k_m * k_m), ((t_m * t_m) * 2.0)) * (k_m * k_m)) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else if (t_m <= 6.5e+86) {
		tmp = 2.0 / ((((k_m / l_m) * (pow(t_m, 3.0) / l_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.6e-105)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k_m * k_m), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	elseif (t_m <= 6.5e+86)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l_m) * Float64((t_m ^ 3.0) / l_m)) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-105], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+86], N[(2.0 / N[(N[(N[(N[(k$95$m / l$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.59999999999999991e-105

   1. Initial program 41.9%

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 72.2%

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

   9. Applied rewrites 82.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 50.3%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}} \]

   if 1.59999999999999991e-105 < t < 6.49999999999999996e86

   1. Initial program 61.2%

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

   3. Taylor expanded in t around 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 43.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 44.2%

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

   10. Applied rewrites 63.6%

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

   if 6.49999999999999996e86 < t

   1. Initial program 73.1%

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 86.4%

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

   9. Applied rewrites 91.2%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 91.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, -0.6666666666666666, 1\right), k\_m \cdot k\_m, \left(t\_m \cdot t\_m\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot l\_m} \cdot \frac{t\_m}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e-27)
    (/
     2.0
     (*
      (/
       (*
        (fma
         (fma (* t_m t_m) -0.6666666666666666 1.0)
         (* k_m k_m)
         (* (* t_m t_m) 2.0))
        (* k_m k_m))
       (* (cos k_m) l_m))
      (/ t_m l_m)))
    (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 2.1e-27) {
		tmp = 2.0 / (((fma(fma((t_m * t_m), -0.6666666666666666, 1.0), (k_m * k_m), ((t_m * t_m) * 2.0)) * (k_m * k_m)) / (cos(k_m) * l_m)) * (t_m / l_m));
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 2.1e-27)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(Float64(t_m * t_m), -0.6666666666666666, 1.0), Float64(k_m * k_m), Float64(Float64(t_m * t_m) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * l_m)) * Float64(t_m / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e-27], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.10000000000000015e-27

   1. Initial program 43.1%

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 50.8%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}} \]

   if 2.10000000000000015e-27 < t

   1. Initial program 71.4%

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 78.8%

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

   9. Applied rewrites 85.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.18 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot \left(k\_m \cdot k\_m\right)\right) \cdot k\_m\right) \cdot k\_m}\\ \mathbf{elif}\;t\_m \leq 1.26 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{\frac{{t\_m}^{3}}{l\_m}}{l\_m}\right) \cdot \left(\left(1 + \frac{k\_m \cdot k\_m}{t\_m \cdot t\_m}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.18e-105)
    (/ 2.0 (* (* (* (/ (/ t_m l_m) l_m) (* k_m k_m)) k_m) k_m))
    (if (<= t_m 1.26e-18)
      (/
       2.0
       (*
        (* (* k_m k_m) (/ (/ (pow t_m 3.0) l_m) l_m))
        (+ (+ 1.0 (/ (* k_m k_m) (* t_m t_m))) 1.0)))
      (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.18e-105) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * (k_m * k_m)) * k_m) * k_m);
	} else if (t_m <= 1.26e-18) {
		tmp = 2.0 / (((k_m * k_m) * ((pow(t_m, 3.0) / l_m) / l_m)) * ((1.0 + ((k_m * k_m) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.18d-105) then
        tmp = 2.0d0 / (((((t_m / l_m) / l_m) * (k_m * k_m)) * k_m) * k_m)
    else if (t_m <= 1.26d-18) then
        tmp = 2.0d0 / (((k_m * k_m) * (((t_m ** 3.0d0) / l_m) / l_m)) * ((1.0d0 + ((k_m * k_m) / (t_m * t_m))) + 1.0d0))
    else
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 1.18e-105) {
		tmp = 2.0 / (((((t_m / l_m) / l_m) * (k_m * k_m)) * k_m) * k_m);
	} else if (t_m <= 1.26e-18) {
		tmp = 2.0 / (((k_m * k_m) * ((Math.pow(t_m, 3.0) / l_m) / l_m)) * ((1.0 + ((k_m * k_m) / (t_m * t_m))) + 1.0));
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 1.18e-105:
		tmp = 2.0 / (((((t_m / l_m) / l_m) * (k_m * k_m)) * k_m) * k_m)
	elif t_m <= 1.26e-18:
		tmp = 2.0 / (((k_m * k_m) * ((math.pow(t_m, 3.0) / l_m) / l_m)) * ((1.0 + ((k_m * k_m) / (t_m * t_m))) + 1.0))
	else:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 1.18e-105)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l_m) / l_m) * Float64(k_m * k_m)) * k_m) * k_m));
	elseif (t_m <= 1.26e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64((t_m ^ 3.0) / l_m) / l_m)) * Float64(Float64(1.0 + Float64(Float64(k_m * k_m) / Float64(t_m * t_m))) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 1.18e-105)
		tmp = 2.0 / (((((t_m / l_m) / l_m) * (k_m * k_m)) * k_m) * k_m);
	elseif (t_m <= 1.26e-18)
		tmp = 2.0 / (((k_m * k_m) * (((t_m ^ 3.0) / l_m) / l_m)) * ((1.0 + ((k_m * k_m) / (t_m * t_m))) + 1.0));
	else
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.18e-105], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.26e-18], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(k$95$m * k$95$m), $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[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.1799999999999999e-105

   1. Initial program 41.9%

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 31.0%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 48.5%

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

   13. Applied rewrites 52.1%

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

   if 1.1799999999999999e-105 < t < 1.26000000000000004e-18

   1. Initial program 55.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 60.4%

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

   if 1.26000000000000004e-18 < 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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 85.9%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.9%

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

Alternative 16: 74.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \left(k\_m \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right)\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\_m\right)}^{2}}{l\_m} \cdot 2\right) \cdot \frac{t\_m}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-30)
    (/ 2.0 (* (* k_m (* k_m (/ (/ t_m l_m) l_m))) (* k_m k_m)))
    (/ 2.0 (* (* (/ (pow (* k_m t_m) 2.0) l_m) 2.0) (/ t_m l_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 4.5e-30) {
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m));
	} else {
		tmp = 2.0 / (((pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4.5d-30) then
        tmp = 2.0d0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m))
    else
        tmp = 2.0d0 / (((((k_m * t_m) ** 2.0d0) / l_m) * 2.0d0) * (t_m / l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 4.5e-30) {
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m));
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 4.5e-30:
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m))
	else:
		tmp = 2.0 / (((math.pow((k_m * t_m), 2.0) / l_m) * 2.0) * (t_m / l_m))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 4.5e-30)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64(t_m / l_m) / l_m))) * Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t_m) ^ 2.0) / l_m) * 2.0) * Float64(t_m / l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 4.5e-30)
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m));
	else
		tmp = 2.0 / (((((k_m * t_m) ^ 2.0) / l_m) * 2.0) * (t_m / l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-30], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.49999999999999967e-30

   1. Initial program 43.1%

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 32.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 47.8%

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

   13. Applied rewrites 51.0%

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

   if 4.49999999999999967e-30 < t

   1. Initial program 71.4%

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 78.8%

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

   9. Applied rewrites 85.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.6%

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

Alternative 17: 68.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \left(k\_m \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}\right)\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot l\_m}{{\left(k\_m \cdot t\_m\right)}^{2} \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e+20)
    (/ 2.0 (* (* k_m (* k_m (/ (/ t_m l_m) l_m))) (* k_m k_m)))
    (/ (* l_m l_m) (* (pow (* k_m t_m) 2.0) t_m)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e+20) {
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m));
	} else {
		tmp = (l_m * l_m) / (pow((k_m * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5d+20) then
        tmp = 2.0d0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m))
    else
        tmp = (l_m * l_m) / (((k_m * t_m) ** 2.0d0) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 5e+20) {
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m));
	} else {
		tmp = (l_m * l_m) / (Math.pow((k_m * t_m), 2.0) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 5e+20:
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m))
	else:
		tmp = (l_m * l_m) / (math.pow((k_m * t_m), 2.0) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 5e+20)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64(Float64(t_m / l_m) / l_m))) * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l_m * l_m) / Float64((Float64(k_m * t_m) ^ 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 5e+20)
		tmp = 2.0 / ((k_m * (k_m * ((t_m / l_m) / l_m))) * (k_m * k_m));
	else
		tmp = (l_m * l_m) / (((k_m * t_m) ^ 2.0) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e+20], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Power[N[(k$95$m * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5e20

   1. Initial program 44.3%

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 33.1%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 47.3%

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

   13. Applied rewrites 50.4%

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

   if 5e20 < t

   1. Initial program 72.8%

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 83.8%

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

Alternative 18: 66.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{l\_m}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot \left(k\_m \cdot t\_2\right)\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot t\_2\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (/ (/ t_m l_m) l_m)))
   (*
    t_s
    (if (<= t_m 9.8e-26)
      (/ 2.0 (* (* k_m (* k_m t_2)) (* k_m k_m)))
      (/ 2.0 (* (* (* (* t_m t_2) 2.0) (* k_m k_m)) t_m))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = (t_m / l_m) / l_m;
	double tmp;
	if (t_m <= 9.8e-26) {
		tmp = 2.0 / ((k_m * (k_m * t_2)) * (k_m * k_m));
	} else {
		tmp = 2.0 / ((((t_m * t_2) * 2.0) * (k_m * k_m)) * t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l_m) / l_m
    if (t_m <= 9.8d-26) then
        tmp = 2.0d0 / ((k_m * (k_m * t_2)) * (k_m * k_m))
    else
        tmp = 2.0d0 / ((((t_m * t_2) * 2.0d0) * (k_m * k_m)) * t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = (t_m / l_m) / l_m;
	double tmp;
	if (t_m <= 9.8e-26) {
		tmp = 2.0 / ((k_m * (k_m * t_2)) * (k_m * k_m));
	} else {
		tmp = 2.0 / ((((t_m * t_2) * 2.0) * (k_m * k_m)) * t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	t_2 = (t_m / l_m) / l_m
	tmp = 0
	if t_m <= 9.8e-26:
		tmp = 2.0 / ((k_m * (k_m * t_2)) * (k_m * k_m))
	else:
		tmp = 2.0 / ((((t_m * t_2) * 2.0) * (k_m * k_m)) * t_m)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(Float64(t_m / l_m) / l_m)
	tmp = 0.0
	if (t_m <= 9.8e-26)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * t_2)) * Float64(k_m * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_2) * 2.0) * Float64(k_m * k_m)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	t_2 = (t_m / l_m) / l_m;
	tmp = 0.0;
	if (t_m <= 9.8e-26)
		tmp = 2.0 / ((k_m * (k_m * t_2)) * (k_m * k_m));
	else
		tmp = 2.0 / ((((t_m * t_2) * 2.0) * (k_m * k_m)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.8e-26], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.7999999999999998e-26

   1. Initial program 43.2%

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 32.5%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 47.9%

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

   13. Applied rewrites 51.1%

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

   if 9.7999999999999998e-26 < t

   1. Initial program 72.0%

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

   5. Applied rewrites 78.6%

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

   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

   8. Applied rewrites 14.5%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 69.6%

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

Alternative 19: 59.4% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

5. Applied rewrites 72.6%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 28.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) \cdot t}{\ell \cdot \ell} - \left(-\frac{\frac{{t}^{3}}{\ell}}{\ell}\right), k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

9. Taylor expanded in t around 0

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

11. Applied rewrites 50.6%

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

13. Applied rewrites 53.0%

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

Alternative 20: 51.0% accurate, 12.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(t_s, t_m, l_m, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m * l_m) / ((k_m * k_m) * ((t_m * t_m) * t_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 42.4%

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

7. Applied rewrites 42.4%

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

Reproduce

herbie shell --seed 2025025 
(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))))