Toniolo and Linder, Equation (10+)

Percentage Accurate: 56.0% → 92.2%

Time: 13.3s

Alternatives: 24

Speedup: 9.4×

• 27.9% 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

real(8) function code(t, l, k)
    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: 56.0% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\_m\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot {\left(\frac{\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}{2}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.2e-122)
    (/ (cos k) (* (* k t_m) (* (/ (pow (sin k) 2.0) l) (/ k (* 2.0 l)))))
    (*
     (/ l t_m)
     (pow
      (/
       (*
        (+ (pow (/ k t_m) 2.0) 2.0)
        (* (* (tan k) t_m) (/ (* (sin k) t_m) l)))
       2.0)
      -1.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-122) {
		tmp = cos(k) / ((k * t_m) * ((pow(sin(k), 2.0) / l) * (k / (2.0 * l))));
	} else {
		tmp = (l / t_m) * pow((((pow((k / t_m), 2.0) + 2.0) * ((tan(k) * t_m) * ((sin(k) * t_m) / l))) / 2.0), -1.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.2d-122) then
        tmp = cos(k) / ((k * t_m) * (((sin(k) ** 2.0d0) / l) * (k / (2.0d0 * l))))
    else
        tmp = (l / t_m) * ((((((k / t_m) ** 2.0d0) + 2.0d0) * ((tan(k) * t_m) * ((sin(k) * t_m) / l))) / 2.0d0) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-122) {
		tmp = Math.cos(k) / ((k * t_m) * ((Math.pow(Math.sin(k), 2.0) / l) * (k / (2.0 * l))));
	} else {
		tmp = (l / t_m) * Math.pow((((Math.pow((k / t_m), 2.0) + 2.0) * ((Math.tan(k) * t_m) * ((Math.sin(k) * t_m) / l))) / 2.0), -1.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.2e-122:
		tmp = math.cos(k) / ((k * t_m) * ((math.pow(math.sin(k), 2.0) / l) * (k / (2.0 * l))))
	else:
		tmp = (l / t_m) * math.pow((((math.pow((k / t_m), 2.0) + 2.0) * ((math.tan(k) * t_m) * ((math.sin(k) * t_m) / l))) / 2.0), -1.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.2e-122)
		tmp = Float64(cos(k) / Float64(Float64(k * t_m) * Float64(Float64((sin(k) ^ 2.0) / l) * Float64(k / Float64(2.0 * l)))));
	else
		tmp = Float64(Float64(l / t_m) * (Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l))) / 2.0) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.2e-122)
		tmp = cos(k) / ((k * t_m) * (((sin(k) ^ 2.0) / l) * (k / (2.0 * l))));
	else
		tmp = (l / t_m) * ((((((k / t_m) ^ 2.0) + 2.0) * ((tan(k) * t_m) * ((sin(k) * t_m) / l))) / 2.0) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-122], N[(N[Cos[k], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2e-122

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 77.5%

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

   if 2.2e-122 < t

   1. Initial program 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. unpow-prod-down N/A

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

   10. Applied rewrites 95.7%

       \[\leadsto \color{blue}{\frac{\ell}{t} \cdot {\left(\frac{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right)}{2}\right)}^{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot {\left(\frac{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right)}{2}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\_m\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-128)
    (/ (cos k) (* (* k t_m) (* (/ (pow (sin k) 2.0) l) (/ k (* 2.0 l)))))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ (/ k t_m) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-128) {
		tmp = cos(k) / ((k * t_m) * ((pow(sin(k), 2.0) / l) * (k / (2.0 * l))));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, ((k / t_m) / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-128)
		tmp = Float64(cos(k) / Float64(Float64(k * t_m) * Float64(Float64((sin(k) ^ 2.0) / l) * Float64(k / Float64(2.0 * l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-128], N[(N[Cos[k], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.19999999999999961e-128

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 77.5%

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

   if 5.19999999999999961e-128 < t

   1. Initial program 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.8

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

   10. Applied rewrites 93.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t}}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-128}:\\ \;\;\;\;\left(\ell \cdot \left(\ell \cdot \frac{2}{{\sin k}^{2} \cdot k}\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-128)
    (* (* l (* l (/ 2.0 (* (pow (sin k) 2.0) k)))) (/ (/ (cos k) t_m) k))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ (/ k t_m) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-128) {
		tmp = (l * (l * (2.0 / (pow(sin(k), 2.0) * k)))) * ((cos(k) / t_m) / k);
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, ((k / t_m) / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-128)
		tmp = Float64(Float64(l * Float64(l * Float64(2.0 / Float64((sin(k) ^ 2.0) * k)))) * Float64(Float64(cos(k) / t_m) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-128], N[(N[(l * N[(l * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.19999999999999961e-128

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 77.9%

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

   if 5.19999999999999961e-128 < t

   1. Initial program 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.8

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

   10. Applied rewrites 93.8%

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

Alternative 4: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot {\left(\frac{\ell}{\sin k}\right)}^{2}\right) \cdot \cos k}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-128)
    (/ (* (* (/ 2.0 k) (pow (/ l (sin k)) 2.0)) (cos k)) (* k t_m))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ (/ k t_m) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-128) {
		tmp = (((2.0 / k) * pow((l / sin(k)), 2.0)) * cos(k)) / (k * t_m);
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, ((k / t_m) / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-128)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) * (Float64(l / sin(k)) ^ 2.0)) * cos(k)) / Float64(k * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-128], N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.19999999999999961e-128

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 73.5%

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

   if 5.19999999999999961e-128 < t

   1. Initial program 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.3

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

   4. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 93.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.8

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

   10. Applied rewrites 93.8%

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

Alternative 5: 86.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.65 \cdot 10^{-145}:\\ \;\;\;\;\frac{\cos k \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)}{\left(k \cdot t\_m\right) \cdot \left({\sin k}^{2} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.65e-145)
    (/ (* (cos k) (* (* l l) 2.0)) (* (* k t_m) (* (pow (sin k) 2.0) k)))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ (/ k t_m) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.65e-145) {
		tmp = (cos(k) * ((l * l) * 2.0)) / ((k * t_m) * (pow(sin(k), 2.0) * k));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, ((k / t_m) / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.65e-145)
		tmp = Float64(Float64(cos(k) * Float64(Float64(l * l) * 2.0)) / Float64(Float64(k * t_m) * Float64((sin(k) ^ 2.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.65e-145], N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.65e-145

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.6%

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

   9. Applied rewrites 69.2%

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

   if 2.65e-145 < t

   1. Initial program 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 94.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 94.0

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

   8. Applied rewrites 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 94.0

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

   10. Applied rewrites 94.0%

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

Alternative 6: 86.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.65 \cdot 10^{-145}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.65e-145)
    (* (* 2.0 (* l l)) (/ (cos k) (* (* (* (pow (sin k) 2.0) t_m) k) k)))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ (/ k t_m) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.65e-145) {
		tmp = (2.0 * (l * l)) * (cos(k) / (((pow(sin(k), 2.0) * t_m) * k) * k));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, ((k / t_m) / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.65e-145)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.65e-145], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.65e-145

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.8%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 68.8%

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

   if 2.65e-145 < t

   1. Initial program 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 94.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 94.0

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

   8. Applied rewrites 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 94.0

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

   10. Applied rewrites 94.0%

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

Alternative 7: 86.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.65 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k} \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.65e-145)
    (*
     (/ (* (* l l) 2.0) (* (- 0.5 (* 0.5 (cos (+ k k)))) k))
     (/ (/ (cos k) t_m) k))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ (/ k t_m) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.65e-145) {
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k);
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, ((k / t_m) / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.65e-145)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * k)) * Float64(Float64(cos(k) / t_m) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(Float64(k / t_m) / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.65e-145], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.65e-145

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.6%

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

   9. Applied rewrites 67.4%

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

   if 2.65e-145 < t

   1. Initial program 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 94.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 94.0

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

   8. Applied rewrites 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 94.0

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

   10. Applied rewrites 94.0%

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

Alternative 8: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k} \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-145)
    (*
     (/ (* (* l l) 2.0) (* (- 0.5 (* 0.5 (cos (+ k k)))) k))
     (/ (/ (cos k) t_m) k))
    (if (<= t_m 5.6e+102)
      (/
       2.0
       (*
        (* (* (/ (* t_m t_m) l) (* (/ t_m l) k)) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (/
       2.0
       (* (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-145) {
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k);
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / (((((t_m * t_m) / l) * ((t_m / l) * k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.35d-145) then
        tmp = (((l * l) * 2.0d0) / ((0.5d0 - (0.5d0 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k)
    else if (t_m <= 5.6d+102) then
        tmp = 2.0d0 / (((((t_m * t_m) / l) * ((t_m / l) * k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-145) {
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * Math.cos((k + k)))) * k)) * ((Math.cos(k) / t_m) / k);
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / (((((t_m * t_m) / l) * ((t_m / l) * k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((Math.sin(k) * t_m) / l)) * Math.tan(k))) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.35e-145:
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * math.cos((k + k)))) * k)) * ((math.cos(k) / t_m) / k)
	elif t_m <= 5.6e+102:
		tmp = 2.0 / (((((t_m * t_m) / l) * ((t_m / l) * k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((t_m / l) * ((t_m * ((math.sin(k) * t_m) / l)) * math.tan(k))) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.35e-145)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * k)) * Float64(Float64(cos(k) / t_m) / k));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(Float64(t_m / l) * k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.35e-145)
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k);
	elseif (t_m <= 5.6e+102)
		tmp = 2.0 / (((((t_m * t_m) / l) * ((t_m / l) * k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-145], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-145

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.6%

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

   9. Applied rewrites 67.4%

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

   if 2.3500000000000001e-145 < t < 5.60000000000000037e102

   1. Initial program 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if 5.60000000000000037e102 < t

   1. Initial program 53.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.0

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

   4. Applied rewrites 37.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.2%

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

Alternative 9: 85.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.65 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k} \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.65e-145)
    (*
     (/ (* (* l l) 2.0) (* (- 0.5 (* 0.5 (cos (+ k k)))) k))
     (/ (/ (cos k) t_m) k))
    (/
     2.0
     (*
      (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k)))
      (fma k (/ k (* t_m t_m)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.65e-145) {
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k);
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * fma(k, (k / (t_m * t_m)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.65e-145)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * k)) * Float64(Float64(cos(k) / t_m) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.65e-145], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.65e-145

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.6%

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

   9. Applied rewrites 67.4%

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

   if 2.65e-145 < t

   1. Initial program 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 94.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 94.0

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

   8. Applied rewrites 94.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 94.0

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

   10. Applied rewrites 94.0%

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

Alternative 10: 82.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-145}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k} \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{elif}\;t\_m \leq 1.95 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (/ k l))))
   (*
    t_s
    (if (<= t_m 2.35e-145)
      (*
       (/ (* (* l l) 2.0) (* (- 0.5 (* 0.5 (cos (+ k k)))) k))
       (/ (/ (cos k) t_m) k))
      (if (<= t_m 1.95e+99)
        (/ 2.0 (* (* t_m (* t_2 t_2)) (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
        (/
         2.0
         (* (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k))) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if (t_m <= 2.35e-145) {
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k);
	} else if (t_m <= 1.95e+99) {
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (k / l)
    if (t_m <= 2.35d-145) then
        tmp = (((l * l) * 2.0d0) / ((0.5d0 - (0.5d0 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k)
    else if (t_m <= 1.95d+99) then
        tmp = 2.0d0 / ((t_m * (t_2 * t_2)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if (t_m <= 2.35e-145) {
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * Math.cos((k + k)))) * k)) * ((Math.cos(k) / t_m) / k);
	} else if (t_m <= 1.95e+99) {
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((Math.sin(k) * t_m) / l)) * Math.tan(k))) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * (k / l)
	tmp = 0
	if t_m <= 2.35e-145:
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * math.cos((k + k)))) * k)) * ((math.cos(k) / t_m) / k)
	elif t_m <= 1.95e+99:
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((t_m / l) * ((t_m * ((math.sin(k) * t_m) / l)) * math.tan(k))) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(k / l))
	tmp = 0.0
	if (t_m <= 2.35e-145)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * k)) * Float64(Float64(cos(k) / t_m) / k));
	elseif (t_m <= 1.95e+99)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (k / l);
	tmp = 0.0;
	if (t_m <= 2.35e-145)
		tmp = (((l * l) * 2.0) / ((0.5 - (0.5 * cos((k + k)))) * k)) * ((cos(k) / t_m) / k);
	elseif (t_m <= 1.95e+99)
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.35e-145], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e+99], N[(2.0 / N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-145

   1. Initial program 49.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 71.6%

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

   9. Applied rewrites 67.4%

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

   if 2.3500000000000001e-145 < t < 1.94999999999999997e99

   1. Initial program 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 38.6

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

   4. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. unswap-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 91.1

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

   9. Applied rewrites 91.1%

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

   if 1.94999999999999997e99 < t

   1. Initial program 53.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.0

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

   4. Applied rewrites 37.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.2%

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

Alternative 11: 78.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 20000000000000:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(t\_2 \cdot t\_2\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot \tan k\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (/ k l))))
   (*
    t_s
    (if (<= (* l l) 20000000000000.0)
      (/
       2.0
       (*
        (* t_m (* (fma (* 0.16666666666666666 k) k 1.0) (* t_2 t_2)))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (/
       2.0
       (* (* t_m (* (/ t_m l) (* (/ (* (sin k) t_m) l) (tan k)))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if ((l * l) <= 20000000000000.0) {
		tmp = 2.0 / ((t_m * (fma((0.16666666666666666 * k), k, 1.0) * (t_2 * t_2))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((t_m * ((t_m / l) * (((sin(k) * t_m) / l) * tan(k)))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(k / l))
	tmp = 0.0
	if (Float64(l * l) <= 20000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(Float64(0.16666666666666666 * k), k, 1.0) * Float64(t_2 * t_2))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * tan(k)))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 20000000000000.0], N[(2.0 / N[(N[(t$95$m * N[(N[(N[(0.16666666666666666 * k), $MachinePrecision] * k + 1.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e13

   1. Initial program 63.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 22.3

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

   4. Applied rewrites 22.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

   6. Applied rewrites 87.8%

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

   7. Taylor expanded in k around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot k}, k, 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \frac{\color{blue}{{t}^{2} \cdot {k}^{2}}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left({t}^{2} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. times-frac N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. unswap-sqr N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Applied rewrites 88.1%

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 2e13 < (*.f64 l l)

   1. Initial program 42.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 9.2

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 9.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot e^{\log t \cdot 3 - \log \ell \cdot 2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 66.1%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 780000000:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(t\_2 \cdot t\_2\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (/ k l))))
   (*
    t_s
    (if (<= l 780000000.0)
      (/
       2.0
       (*
        (* t_m (* (fma (* 0.16666666666666666 k) k 1.0) (* t_2 t_2)))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (/
       2.0
       (* (* (/ t_m l) (* (* t_m (/ (* (sin k) t_m) l)) (tan k))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if (l <= 780000000.0) {
		tmp = 2.0 / ((t_m * (fma((0.16666666666666666 * k), k, 1.0) * (t_2 * t_2))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t_m / l) * ((t_m * ((sin(k) * t_m) / l)) * tan(k))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(k / l))
	tmp = 0.0
	if (l <= 780000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(Float64(0.16666666666666666 * k), k, 1.0) * Float64(t_2 * t_2))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(sin(k) * t_m) / l)) * tan(k))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 780000000.0], N[(2.0 / N[(N[(t$95$m * N[(N[(N[(0.16666666666666666 * k), $MachinePrecision] * k + 1.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.8e8

   1. Initial program 56.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 15.3

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 15.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot e^{\log t \cdot 3 - \log \ell \cdot 2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 80.1%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + \frac{{t}^{2}}{{\ell}^{2}}\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{1}{6}\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}} + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + \color{blue}{\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\frac{1}{6} \cdot \color{blue}{\left(k \cdot k\right)} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\color{blue}{\left(\frac{1}{6} \cdot k\right) \cdot k} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot k}, k, 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \frac{\color{blue}{{t}^{2} \cdot {k}^{2}}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left({t}^{2} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. times-frac N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. unswap-sqr N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Applied rewrites 78.3%

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 7.8e8 < l

   1. Initial program 44.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.8

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot e^{\log t \cdot 3 - \log \ell \cdot 2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 69.2%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right) \cdot t\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)} \cdot t\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right) \cdot t\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{\color{blue}{t \cdot \sin k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{\color{blue}{t \cdot \sin k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 72.8

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. lift-*.f64 72.8

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 72.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right)\right) \cdot \color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right)\right) \cdot \color{blue}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(t\_2 \cdot t\_2\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (/ k l))))
   (*
    t_s
    (if (<= t_m 2.1e-153)
      (*
       (* 2.0 (* l l))
       (/ (/ (cos k) t_m) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) k) k)))
      (/
       2.0
       (*
        (* t_m (* (fma (* 0.16666666666666666 k) k 1.0) (* t_2 t_2)))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if (t_m <= 2.1e-153) {
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * k) * k));
	} else {
		tmp = 2.0 / ((t_m * (fma((0.16666666666666666 * k), k, 1.0) * (t_2 * t_2))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(k / l))
	tmp = 0.0
	if (t_m <= 2.1e-153)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(Float64(cos(k) / t_m) / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k + k)))) * k) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(Float64(0.16666666666666666 * k), k, 1.0) * Float64(t_2 * t_2))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.1e-153], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(N[(0.16666666666666666 * k), $MachinePrecision] * k + 1.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.10000000000000004e-153

   1. Initial program 50.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 63.7

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.7%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k} \]

   if 2.10000000000000004e-153 < t

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 37.7

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 37.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot e^{\log t \cdot 3 - \log \ell \cdot 2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 89.3%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + \frac{{t}^{2}}{{\ell}^{2}}\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{1}{6}\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}} + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + \color{blue}{\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\frac{1}{6} \cdot \color{blue}{\left(k \cdot k\right)} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\color{blue}{\left(\frac{1}{6} \cdot k\right) \cdot k} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot k}, k, 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \frac{\color{blue}{{t}^{2} \cdot {k}^{2}}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left({t}^{2} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. times-frac N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. unswap-sqr N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Applied rewrites 84.5%

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 75.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(t\_2 \cdot t\_2\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (/ k l))))
   (*
    t_s
    (if (<= (* l l) 2e+151)
      (/
       2.0
       (*
        (* t_m (* (fma (* 0.16666666666666666 k) k 1.0) (* t_2 t_2)))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (/ (* (/ l t_m) l) (pow (* k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if ((l * l) <= 2e+151) {
		tmp = 2.0 / ((t_m * (fma((0.16666666666666666 * k), k, 1.0) * (t_2 * t_2))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = ((l / t_m) * l) / pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(k / l))
	tmp = 0.0
	if (Float64(l * l) <= 2e+151)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(Float64(0.16666666666666666 * k), k, 1.0) * Float64(t_2 * t_2))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / (Float64(k * t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e+151], N[(2.0 / N[(N[(t$95$m * N[(N[(N[(0.16666666666666666 * k), $MachinePrecision] * k + 1.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.00000000000000003e151

   1. Initial program 62.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 19.8

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 19.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot e^{\log t \cdot 3 - \log \ell \cdot 2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 86.7%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + \frac{{t}^{2}}{{\ell}^{2}}\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left({k}^{2} \cdot \frac{1}{6}\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}} + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}} + \color{blue}{\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\frac{1}{6} \cdot \color{blue}{\left(k \cdot k\right)} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(\color{blue}{\left(\frac{1}{6} \cdot k\right) \cdot k} + 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right)} \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot k}, k, 1\right) \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \frac{\color{blue}{{t}^{2} \cdot {k}^{2}}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left({t}^{2} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. times-frac N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. unswap-sqr N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot k, k, 1\right) \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Applied rewrites 86.3%

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666 \cdot k, k, 1\right) \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 2.00000000000000003e151 < (*.f64 l l)

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 50.6

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.8%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{\left(k \cdot t\right)}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 75.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-64}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (/ k l))))
   (*
    t_s
    (if (<= (* l l) 1e-64)
      (/ 2.0 (* (* t_m (* t_2 t_2)) (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (/ (* (/ l t_m) l) (pow (* k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if ((l * l) <= 1e-64) {
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = ((l / t_m) * l) / pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (k / l)
    if ((l * l) <= 1d-64) then
        tmp = 2.0d0 / ((t_m * (t_2 * t_2)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = ((l / t_m) * l) / ((k * t_m) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (k / l);
	double tmp;
	if ((l * l) <= 1e-64) {
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = ((l / t_m) * l) / Math.pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * (k / l)
	tmp = 0
	if (l * l) <= 1e-64:
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = ((l / t_m) * l) / math.pow((k * t_m), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(k / l))
	tmp = 0.0
	if (Float64(l * l) <= 1e-64)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / (Float64(k * t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (k / l);
	tmp = 0.0;
	if ((l * l) <= 1e-64)
		tmp = 2.0 / ((t_m * (t_2 * t_2)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = ((l / t_m) * l) / ((k * t_m) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-64], N[(2.0 / N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999965e-65

   1. Initial program 62.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-log.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t} \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-log.f64 22.9

          \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 22.9%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\sin k \cdot e^{\log t \cdot 3 - \log \ell \cdot 2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 89.7%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \frac{\color{blue}{{t}^{2} \cdot {k}^{2}}}{{\ell}^{2}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left({t}^{2} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. unswap-sqr N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\left(t \cdot \frac{k}{\ell}\right)} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\left(t \cdot \color{blue}{\frac{k}{\ell}}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\left(t \cdot \frac{k}{\ell}\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 90.7

          \[\leadsto \frac{2}{\left(t \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \color{blue}{\frac{k}{\ell}}\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. Applied rewrites 90.7%

      \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   if 9.99999999999999965e-65 < (*.f64 l l)

   1. Initial program 46.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 52.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.6%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{\left(k \cdot t\right)}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 72.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot k} \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t\_m}^{-3}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-88)
    (* (/ (* (* l l) 2.0) (* (* k k) k)) (/ (/ (cos k) t_m) k))
    (if (<= t_m 4e+103)
      (/ (* (/ l k) (* l (pow t_m -3.0))) k)
      (/ (* (/ l t_m) l) (pow (* k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-88) {
		tmp = (((l * l) * 2.0) / ((k * k) * k)) * ((cos(k) / t_m) / k);
	} else if (t_m <= 4e+103) {
		tmp = ((l / k) * (l * pow(t_m, -3.0))) / k;
	} else {
		tmp = ((l / t_m) * l) / pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.1d-88) then
        tmp = (((l * l) * 2.0d0) / ((k * k) * k)) * ((cos(k) / t_m) / k)
    else if (t_m <= 4d+103) then
        tmp = ((l / k) * (l * (t_m ** (-3.0d0)))) / k
    else
        tmp = ((l / t_m) * l) / ((k * t_m) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-88) {
		tmp = (((l * l) * 2.0) / ((k * k) * k)) * ((Math.cos(k) / t_m) / k);
	} else if (t_m <= 4e+103) {
		tmp = ((l / k) * (l * Math.pow(t_m, -3.0))) / k;
	} else {
		tmp = ((l / t_m) * l) / Math.pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.1e-88:
		tmp = (((l * l) * 2.0) / ((k * k) * k)) * ((math.cos(k) / t_m) / k)
	elif t_m <= 4e+103:
		tmp = ((l / k) * (l * math.pow(t_m, -3.0))) / k
	else:
		tmp = ((l / t_m) * l) / math.pow((k * t_m), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-88)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k * k) * k)) * Float64(Float64(cos(k) / t_m) / k));
	elseif (t_m <= 4e+103)
		tmp = Float64(Float64(Float64(l / k) * Float64(l * (t_m ^ -3.0))) / k);
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / (Float64(k * t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.1e-88)
		tmp = (((l * l) * 2.0) / ((k * k) * k)) * ((cos(k) / t_m) / k);
	elseif (t_m <= 4e+103)
		tmp = ((l / k) * (l * (t_m ^ -3.0))) / k;
	else
		tmp = ((l / t_m) * l) / ((k * t_m) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-88], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+103], N[(N[(N[(l / k), $MachinePrecision] * N[(l * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.10000000000000002e-88

   1. Initial program 47.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 64.1

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.5%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{\sin k}^{2} \cdot k} \cdot \color{blue}{\frac{\frac{\cos k}{t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{{k}^{2} \cdot k} \cdot \frac{\frac{\cos k}{t}}{k} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.1%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot k} \cdot \frac{\frac{\cos k}{t}}{k} \]

   if 1.10000000000000002e-88 < t < 4e103

   1. Initial program 80.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 81.1

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.7%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-3}\right)}{\color{blue}{k}} \]

   if 4e103 < t

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 51.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.7%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.3%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{\left(k \cdot t\right)}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 71.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t\_m}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t\_m}^{-3}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05e-88)
    (* (/ 2.0 (pow k 4.0)) (/ (* l l) t_m))
    (if (<= t_m 4e+103)
      (/ (* (/ l k) (* l (pow t_m -3.0))) k)
      (/ (* (/ l t_m) l) (pow (* k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-88) {
		tmp = (2.0 / pow(k, 4.0)) * ((l * l) / t_m);
	} else if (t_m <= 4e+103) {
		tmp = ((l / k) * (l * pow(t_m, -3.0))) / k;
	} else {
		tmp = ((l / t_m) * l) / pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.05d-88) then
        tmp = (2.0d0 / (k ** 4.0d0)) * ((l * l) / t_m)
    else if (t_m <= 4d+103) then
        tmp = ((l / k) * (l * (t_m ** (-3.0d0)))) / k
    else
        tmp = ((l / t_m) * l) / ((k * t_m) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-88) {
		tmp = (2.0 / Math.pow(k, 4.0)) * ((l * l) / t_m);
	} else if (t_m <= 4e+103) {
		tmp = ((l / k) * (l * Math.pow(t_m, -3.0))) / k;
	} else {
		tmp = ((l / t_m) * l) / Math.pow((k * t_m), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.05e-88:
		tmp = (2.0 / math.pow(k, 4.0)) * ((l * l) / t_m)
	elif t_m <= 4e+103:
		tmp = ((l / k) * (l * math.pow(t_m, -3.0))) / k
	else:
		tmp = ((l / t_m) * l) / math.pow((k * t_m), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-88)
		tmp = Float64(Float64(2.0 / (k ^ 4.0)) * Float64(Float64(l * l) / t_m));
	elseif (t_m <= 4e+103)
		tmp = Float64(Float64(Float64(l / k) * Float64(l * (t_m ^ -3.0))) / k);
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / (Float64(k * t_m) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.05e-88)
		tmp = (2.0 / (k ^ 4.0)) * ((l * l) / t_m);
	elseif (t_m <= 4e+103)
		tmp = ((l / k) * (l * (t_m ^ -3.0))) / k;
	else
		tmp = ((l / t_m) * l) / ((k * t_m) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05e-88], N[(N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+103], N[(N[(N[(l / k), $MachinePrecision] * N[(l * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.05e-88

   1. Initial program 47.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 64.1

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

   if 1.05e-88 < t < 4e103

   1. Initial program 80.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 81.1

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.7%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-3}\right)}{\color{blue}{k}} \]

   if 4e103 < t

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 51.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.7%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 86.3%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{\left(k \cdot t\right)}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 67.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6400000000000:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{k}}{\left(t\_m \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6400000000000.0)
    (/ (* (/ l t_m) (/ l k)) (* (* t_m t_m) k))
    (* (* (/ (/ -0.16666666666666666 (* k k)) t_m) (* 2.0 l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6400000000000.0) {
		tmp = ((l / t_m) * (l / k)) / ((t_m * t_m) * k);
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6400000000000.0d0) then
        tmp = ((l / t_m) * (l / k)) / ((t_m * t_m) * k)
    else
        tmp = ((((-0.16666666666666666d0) / (k * k)) / t_m) * (2.0d0 * l)) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6400000000000.0) {
		tmp = ((l / t_m) * (l / k)) / ((t_m * t_m) * k);
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6400000000000.0:
		tmp = ((l / t_m) * (l / k)) / ((t_m * t_m) * k)
	else:
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6400000000000.0)
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / k)) / Float64(Float64(t_m * t_m) * k));
	else
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 / Float64(k * k)) / t_m) * Float64(2.0 * l)) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6400000000000.0)
		tmp = ((l / t_m) * (l / k)) / ((t_m * t_m) * k);
	else
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6400000000000.0], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.4e12

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.9%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.1%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot k}} \]

   if 6.4e12 < k

   1. Initial program 47.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 72.9

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 72.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 67.3%

       \[\leadsto \left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 63.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 16500000000000:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 16500000000000.0)
    (* (/ l (* (* t_m t_m) t_m)) (/ (/ l k) k))
    (* (* (/ (/ -0.16666666666666666 (* k k)) t_m) (* 2.0 l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 16500000000000.0) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 16500000000000.0d0) then
        tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
    else
        tmp = ((((-0.16666666666666666d0) / (k * k)) / t_m) * (2.0d0 * l)) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 16500000000000.0) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 16500000000000.0:
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
	else:
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 16500000000000.0)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 / Float64(k * k)) / t_m) * Float64(2.0 * l)) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 16500000000000.0)
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	else
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 16500000000000.0], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.65e13

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

   if 1.65e13 < k

   1. Initial program 47.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 72.9

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 72.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 67.3%

       \[\leadsto \left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 16500000000000:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 16500000000000.0)
    (* (/ l t_m) (/ (/ l (* k k)) (* t_m t_m)))
    (* (* (/ (/ -0.16666666666666666 (* k k)) t_m) (* 2.0 l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 16500000000000.0) {
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * t_m));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 16500000000000.0d0) then
        tmp = (l / t_m) * ((l / (k * k)) / (t_m * t_m))
    else
        tmp = ((((-0.16666666666666666d0) / (k * k)) / t_m) * (2.0d0 * l)) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 16500000000000.0) {
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * t_m));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 16500000000000.0:
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * t_m))
	else:
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 16500000000000.0)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(k * k)) / Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 / Float64(k * k)) / t_m) * Float64(2.0 * l)) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 16500000000000.0)
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * t_m));
	else
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 16500000000000.0], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.65e13

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.1%

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{t \cdot t}} \]

   if 1.65e13 < k

   1. Initial program 47.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 72.9

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 72.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 67.3%

       \[\leadsto \left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 61.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6400000000000:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6400000000000.0)
    (/ (* (/ l t_m) l) (* (* t_m t_m) (* k k)))
    (* (* (/ (/ -0.16666666666666666 (* k k)) t_m) (* 2.0 l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6400000000000.0) {
		tmp = ((l / t_m) * l) / ((t_m * t_m) * (k * k));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6400000000000.0d0) then
        tmp = ((l / t_m) * l) / ((t_m * t_m) * (k * k))
    else
        tmp = ((((-0.16666666666666666d0) / (k * k)) / t_m) * (2.0d0 * l)) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6400000000000.0) {
		tmp = ((l / t_m) * l) / ((t_m * t_m) * (k * k));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6400000000000.0:
		tmp = ((l / t_m) * l) / ((t_m * t_m) * (k * k))
	else:
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6400000000000.0)
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(Float64(t_m * t_m) * Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 / Float64(k * k)) / t_m) * Float64(2.0 * l)) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6400000000000.0)
		tmp = ((l / t_m) * l) / ((t_m * t_m) * (k * k));
	else
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6400000000000.0], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.4e12

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.9%

      \[\leadsto \left({t}^{-1} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.9%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \left(-\ell\right)}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(-k\right) \cdot k\right)}} \]

   if 6.4e12 < k

   1. Initial program 47.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 72.9

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 72.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 67.3%

       \[\leadsto \left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6400000000000:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 16500000000000:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 16500000000000.0)
    (* (/ l (* (* t_m t_m) t_m)) (/ l (* k k)))
    (* (* (/ (/ -0.16666666666666666 (* k k)) t_m) (* 2.0 l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 16500000000000.0) {
		tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 16500000000000.0d0) then
        tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k))
    else
        tmp = ((((-0.16666666666666666d0) / (k * k)) / t_m) * (2.0d0 * l)) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 16500000000000.0) {
		tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 16500000000000.0:
		tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k))
	else:
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 16500000000000.0)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 / Float64(k * k)) / t_m) * Float64(2.0 * l)) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 16500000000000.0)
		tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k));
	else
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 16500000000000.0], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.65e13

   1. Initial program 55.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]

   if 1.65e13 < k

   1. Initial program 47.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 72.9

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 72.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 67.3%

       \[\leadsto \left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+126}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1e+126)
    (* (* 2.0 (* l l)) (/ -0.16666666666666666 (* (* k t_m) k)))
    (* (* (/ (/ -0.16666666666666666 (* k k)) t_m) (* 2.0 l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1e+126) {
		tmp = (2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1d+126) then
        tmp = (2.0d0 * (l * l)) * ((-0.16666666666666666d0) / ((k * t_m) * k))
    else
        tmp = ((((-0.16666666666666666d0) / (k * k)) / t_m) * (2.0d0 * l)) * l
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1e+126) {
		tmp = (2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k));
	} else {
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1e+126:
		tmp = (2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k))
	else:
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1e+126)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(-0.16666666666666666 / Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(Float64(Float64(Float64(-0.16666666666666666 / Float64(k * k)) / t_m) * Float64(2.0 * l)) * l);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1e+126)
		tmp = (2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k));
	else
		tmp = (((-0.16666666666666666 / (k * k)) / t_m) * (2.0 * l)) * l;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1e+126], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 9.99999999999999925e125

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 59.8

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.8%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.6%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]

   12. Taylor expanded in k around inf

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   13. Step-by-step derivation

   14. Applied rewrites 36.3%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]

   if 9.99999999999999925e125 < l

   1. Initial program 35.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

      8. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

      9. associate-/l/ N/A

         \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      12. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      13. *-commutative N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

      14. unpow2 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

      18. lower-pow.f64 N/A

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

      19. lower-sin.f64 58.6

          \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

   5. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.4%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 14.7%

       \[\leadsto \left(\frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 34.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\_m\right) \cdot k}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* 2.0 (* l l)) (/ -0.16666666666666666 (* (* k t_m) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 * (l * l)) * ((-0.16666666666666666d0) / ((k * t_m) * k)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 * Float64(l * l)) * Float64(-0.16666666666666666 / Float64(Float64(k * t_m) * k))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 * (l * l)) * (-0.16666666666666666 / ((k * t_m) * k)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. unpow2 N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. associate-/l/ N/A

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}} \]

   8. associate-/r* N/A

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{{k}^{2}} \]

   9. associate-/l/ N/A

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\color{blue}{\frac{\cos k}{t}}}{{k}^{2} \cdot {\sin k}^{2}} \]

   12. lower-cos.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\color{blue}{\cos k}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

   13. *-commutative N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{{\sin k}^{2} \cdot {k}^{2}}} \]

   14. unpow2 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}} \]

   15. associate-*r* N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   16. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

   17. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\color{blue}{\left({\sin k}^{2} \cdot k\right)} \cdot k} \]

   18. lower-pow.f64 N/A

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\color{blue}{{\sin k}^{2}} \cdot k\right) \cdot k} \]

   19. lower-sin.f64 59.7

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\color{blue}{\sin k}}^{2} \cdot k\right) \cdot k} \]

5. Applied rewrites 59.7%

   \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left({\sin k}^{2} \cdot k\right) \cdot k}} \]

6. Taylor expanded in k around 0

   \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6} \cdot \frac{{k}^{2}}{t} + \frac{1}{t}}{\color{blue}{{k}^{4}}} \]
7. Step-by-step derivation

8. Applied rewrites 35.6%

   \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{k \cdot k}{t}, -0.16666666666666666, \frac{1}{t}\right)}{\color{blue}{{k}^{4}}} \]

9. Taylor expanded in k around inf

   \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
10. Step-by-step derivation

11. Applied rewrites 32.2%

    \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-0.16666666666666666}{k \cdot k}}{t} \]

12. Taylor expanded in k around inf

    \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{-1}{6}}{{k}^{2} \cdot \color{blue}{t}} \]
13. Step-by-step derivation

14. Applied rewrites 32.8%

    \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024316 
(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))))