Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.0% → 93.7%

Time: 13.8s

Alternatives: 27

Speedup: 9.4×

• 27.4% 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: 54.0% 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: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-74}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\_m\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-74)
    (/ (cos k) (* (* k t_m) (* (/ (pow (sin k) 2.0) l) (/ k (* 2.0 l)))))
    (/
     2.0
     (*
      (* (/ (* (sin k) t_m) l) t_m)
      (* (/ t_m l) (* (tan k) (+ (pow (/ k t_m) 2.0) 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 <= 1.85e-74) {
		tmp = cos(k) / ((k * t_m) * ((pow(sin(k), 2.0) / l) * (k / (2.0 * l))));
	} else {
		tmp = 2.0 / ((((sin(k) * t_m) / l) * t_m) * ((t_m / l) * (tan(k) * (pow((k / t_m), 2.0) + 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 <= 1.85d-74) then
        tmp = cos(k) / ((k * t_m) * (((sin(k) ** 2.0d0) / l) * (k / (2.0d0 * l))))
    else
        tmp = 2.0d0 / ((((sin(k) * t_m) / l) * t_m) * ((t_m / l) * (tan(k) * (((k / t_m) ** 2.0d0) + 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 <= 1.85e-74) {
		tmp = Math.cos(k) / ((k * t_m) * ((Math.pow(Math.sin(k), 2.0) / l) * (k / (2.0 * l))));
	} else {
		tmp = 2.0 / ((((Math.sin(k) * t_m) / l) * t_m) * ((t_m / l) * (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 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 <= 1.85e-74:
		tmp = math.cos(k) / ((k * t_m) * ((math.pow(math.sin(k), 2.0) / l) * (k / (2.0 * l))))
	else:
		tmp = 2.0 / ((((math.sin(k) * t_m) / l) * t_m) * ((t_m / l) * (math.tan(k) * (math.pow((k / t_m), 2.0) + 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 <= 1.85e-74)
		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(Float64(sin(k) * t_m) / l) * t_m) * Float64(Float64(t_m / l) * Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 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 <= 1.85e-74)
		tmp = cos(k) / ((k * t_m) * (((sin(k) ^ 2.0) / l) * (k / (2.0 * l))));
	else
		tmp = 2.0 / ((((sin(k) * t_m) / l) * t_m) * ((t_m / l) * (tan(k) * (((k / t_m) ^ 2.0) + 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, 1.85e-74], 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[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.84999999999999997e-74

   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 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 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 64.9%

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

   9. Applied rewrites 75.0%

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

   if 1.84999999999999997e-74 < t

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      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. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.9

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

   6. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   8. Applied rewrites 97.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-74}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\_m\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1.05e-42)
    (/ (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 <= 1.05e-42) {
		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 <= 1.05e-42)
		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(Float64(t_m / l) * 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, 1.05e-42], 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[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.05000000000000003e-42

   1. Initial program 48.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 65.6%

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

   9. Applied rewrites 75.7%

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

   if 1.05000000000000003e-42 < t

   1. Initial program 65.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.3

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

   8. Applied rewrites 93.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot t\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t}}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{{\sin k}^{2} \cdot k}\right) \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1.05e-42)
    (* (* (* 2.0 l) (/ l (* (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 <= 1.05e-42) {
		tmp = ((2.0 * l) * (l / (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 <= 1.05e-42)
		tmp = Float64(Float64(Float64(2.0 * l) * Float64(l / Float64((sin(k) ^ 2.0) * k))) * Float64(Float64(cos(k) / t_m) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * 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, 1.05e-42], N[(N[(N[(2.0 * l), $MachinePrecision] * N[(l / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.05000000000000003e-42

   1. Initial program 48.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 65.6%

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

   9. Applied rewrites 74.5%

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

   if 1.05000000000000003e-42 < t

   1. Initial program 65.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.3

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

   8. Applied rewrites 93.3%

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

Alternative 4: 90.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\left(\ell \cdot \left(\ell \cdot \frac{2}{{\sin k}^{2} \cdot k}\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1.05e-42)
    (* (* 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 <= 1.05e-42) {
		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 <= 1.05e-42)
		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(Float64(t_m / l) * 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, 1.05e-42], 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[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.05000000000000003e-42

   1. Initial program 48.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 65.6%

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

   9. Applied rewrites 74.2%

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

   if 1.05000000000000003e-42 < t

   1. Initial program 65.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.3

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

   8. Applied rewrites 93.3%

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

Alternative 5: 88.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-48}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot {\left(\frac{\ell}{\sin k}\right)}^{2}\right) \cdot \cos k}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1e-48)
    (/ (* (* (/ 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 <= 1e-48) {
		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 <= 1e-48)
		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(Float64(t_m / l) * 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, 1e-48], 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[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 9.9999999999999997e-49

   1. Initial program 48.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.3

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

   5. Applied rewrites 60.3%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 67.1%

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

   if 9.9999999999999997e-49 < t

   1. Initial program 64.4%

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

      1. lift-*.f64 N/A

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

      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. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 6: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(k \cdot t\_m\right) \cdot k} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 9e-49)
    (* (/ 2.0 (* (* k t_m) k)) (/ (* (* (cos k) l) l) (pow (sin k) 2.0)))
    (/
     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 <= 9e-49) {
		tmp = (2.0 / ((k * t_m) * k)) * (((cos(k) * l) * l) / pow(sin(k), 2.0));
	} 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 <= 9e-49)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * t_m) * k)) * Float64(Float64(Float64(cos(k) * l) * l) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * 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, 9e-49], N[(N[(2.0 / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 9.0000000000000004e-49

   1. Initial program 48.7%

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

      1. lift-*.f64 N/A

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

      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. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 56.1

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

   4. Applied rewrites 56.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 66.1

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

   6. Applied rewrites 66.1%

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

   7. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-sin.f64 66.6

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

   9. Applied rewrites 66.6%

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

   if 9.0000000000000004e-49 < t

   1. Initial program 64.4%

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

      1. lift-*.f64 N/A

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

      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. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.2

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

   8. Applied rewrites 92.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\left(k \cdot t\right) \cdot k} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t}}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\sin k}^{2} \cdot k} \cdot \frac{{k}^{-1}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t\_m}^{-2}\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-81)
    (* (/ (* (* l l) 2.0) (* (pow (sin k) 2.0) k)) (/ (pow k -1.0) t_m))
    (/ (* (/ l k) (* l (pow t_m -2.0))) (* k t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-81) {
		tmp = (((l * l) * 2.0) / (pow(sin(k), 2.0) * k)) * (pow(k, -1.0) / t_m);
	} else {
		tmp = ((l / k) * (l * pow(t_m, -2.0))) / (k * t_m);
	}
	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 <= 4.2d-81) then
        tmp = (((l * l) * 2.0d0) / ((sin(k) ** 2.0d0) * k)) * ((k ** (-1.0d0)) / t_m)
    else
        tmp = ((l / k) * (l * (t_m ** (-2.0d0)))) / (k * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-81) {
		tmp = (((l * l) * 2.0) / (Math.pow(Math.sin(k), 2.0) * k)) * (Math.pow(k, -1.0) / t_m);
	} else {
		tmp = ((l / k) * (l * Math.pow(t_m, -2.0))) / (k * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-81:
		tmp = (((l * l) * 2.0) / (math.pow(math.sin(k), 2.0) * k)) * (math.pow(k, -1.0) / t_m)
	else:
		tmp = ((l / k) * (l * math.pow(t_m, -2.0))) / (k * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-81)
		tmp = Float64(Float64(Float64(Float64(l * l) * 2.0) / Float64((sin(k) ^ 2.0) * k)) * Float64((k ^ -1.0) / t_m));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l * (t_m ^ -2.0))) / Float64(k * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-81)
		tmp = (((l * l) * 2.0) / ((sin(k) ^ 2.0) * k)) * ((k ^ -1.0) / t_m);
	else
		tmp = ((l / k) * (l * (t_m ^ -2.0))) / (k * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-81], N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, -1.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l * N[Power[t$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.1999999999999998e-81

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 53.0%

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

   if 4.1999999999999998e-81 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 75.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{{\sin k}^{2} \cdot k} \cdot \frac{{k}^{-1}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-2}\right)}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\cos k \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)}{\left(k \cdot t\_m\right) \cdot \left({\sin k}^{2} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 6.2e-93)
    (/ (* (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 <= 6.2e-93) {
		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 <= 6.2e-93)
		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(Float64(t_m / l) * 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, 6.2e-93], 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[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 6.19999999999999999e-93

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 64.0%

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

   if 6.19999999999999999e-93 < t

   1. Initial program 66.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 9: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-49}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 9e-49)
    (* (* 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 <= 9e-49) {
		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 <= 9e-49)
		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(Float64(t_m / l) * 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, 9e-49], 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[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 9.0000000000000004e-49

   1. Initial program 48.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 65.5%

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

   if 9.0000000000000004e-49 < t

   1. Initial program 64.4%

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

      1. lift-*.f64 N/A

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

      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. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.2

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

   8. Applied rewrites 92.2%

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

Alternative 10: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-93}:\\ \;\;\;\;\left(\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{t\_m}\right)\right) \cdot {\left(\sin k \cdot k\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 6.2e-93)
    (* (* (* 2.0 l) (* l (/ (cos k) t_m))) (pow (* (sin k) k) -2.0))
    (/
     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 <= 6.2e-93) {
		tmp = ((2.0 * l) * (l * (cos(k) / t_m))) * pow((sin(k) * k), -2.0);
	} 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 <= 6.2e-93)
		tmp = Float64(Float64(Float64(2.0 * l) * Float64(l * Float64(cos(k) / t_m))) * (Float64(sin(k) * k) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * 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, 6.2e-93], N[(N[(N[(2.0 * l), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 6.19999999999999999e-93

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 62.6%

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

   if 6.19999999999999999e-93 < t

   1. Initial program 66.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 11: 82.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-102}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k \cdot {\left(\sin k \cdot k\right)}^{-2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 3.1e-102)
    (* (* 2.0 (* l l)) (/ (* (cos k) (pow (* (sin k) k) -2.0)) 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 <= 3.1e-102) {
		tmp = (2.0 * (l * l)) * ((cos(k) * pow((sin(k) * k), -2.0)) / 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 <= 3.1e-102)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(Float64(cos(k) * (Float64(sin(k) * k) ^ -2.0)) / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * 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, 3.1e-102], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 3.10000000000000013e-102

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 59.2%

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

   if 3.10000000000000013e-102 < t

   1. Initial program 66.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 12: 82.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-102}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{\left(k \cdot \sin k\right)}^{2} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 3.1e-102)
    (* (* 2.0 (* l l)) (/ (cos k) (* (pow (* k (sin k)) 2.0) 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 <= 3.1e-102) {
		tmp = (2.0 * (l * l)) * (cos(k) / (pow((k * sin(k)), 2.0) * 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 <= 3.1e-102)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64((Float64(k * sin(k)) ^ 2.0) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * 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, 3.1e-102], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 3.10000000000000013e-102

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 59.3%

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

   if 3.10000000000000013e-102 < t

   1. Initial program 66.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 13: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1.15e-104)
    (*
     (* 2.0 (* l l))
     (/ (/ (cos k) t_m) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) 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 <= 1.15e-104) {
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * 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 <= 1.15e-104)
		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(Float64(Float64(t_m / l) * 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, 1.15e-104], 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[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.15e-104

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 57.3%

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

   if 1.15e-104 < t

   1. Initial program 66.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 14: 81.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1.15e-104)
    (*
     (* 2.0 (* l l))
     (/ (/ (cos k) t_m) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) 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 <= 1.15e-104) {
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * 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 <= 1.15e-104)
		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(Float64(Float64(t_m / l) * 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, 1.15e-104], 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[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $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 < 1.15e-104

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 57.3%

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

   if 1.15e-104 < t

   1. Initial program 66.2%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 91.0

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

   8. Applied rewrites 91.0%

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

Alternative 15: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{{t\_m}^{3}}}{k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{\sin k \cdot t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 2.1e-40)
    (*
     (* 2.0 (* l l))
     (/ (/ (cos k) t_m) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) k) k)))
    (if (<= t_m 1.85e+22)
      (* (/ (/ l (pow t_m 3.0)) k) (/ l k))
      (/
       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 <= 2.1e-40) {
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * k) * k));
	} else if (t_m <= 1.85e+22) {
		tmp = ((l / pow(t_m, 3.0)) / k) * (l / k);
	} 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 <= 2.1d-40) then
        tmp = (2.0d0 * (l * l)) * ((cos(k) / t_m) / (((0.5d0 - (0.5d0 * cos((k + k)))) * k) * k))
    else if (t_m <= 1.85d+22) then
        tmp = ((l / (t_m ** 3.0d0)) / k) * (l / k)
    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 <= 2.1e-40) {
		tmp = (2.0 * (l * l)) * ((Math.cos(k) / t_m) / (((0.5 - (0.5 * Math.cos((k + k)))) * k) * k));
	} else if (t_m <= 1.85e+22) {
		tmp = ((l / Math.pow(t_m, 3.0)) / k) * (l / k);
	} 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 <= 2.1e-40:
		tmp = (2.0 * (l * l)) * ((math.cos(k) / t_m) / (((0.5 - (0.5 * math.cos((k + k)))) * k) * k))
	elif t_m <= 1.85e+22:
		tmp = ((l / math.pow(t_m, 3.0)) / k) * (l / k)
	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 <= 2.1e-40)
		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)));
	elseif (t_m <= 1.85e+22)
		tmp = Float64(Float64(Float64(l / (t_m ^ 3.0)) / k) * Float64(l / k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * 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 <= 2.1e-40)
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * k) * k));
	elseif (t_m <= 1.85e+22)
		tmp = ((l / (t_m ^ 3.0)) / k) * (l / k);
	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, 2.1e-40], 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], If[LessEqual[t$95$m, 1.85e+22], N[(N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.10000000000000018e-40

   1. Initial program 48.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 60.2

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 58.3%

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

   if 2.10000000000000018e-40 < t < 1.8499999999999999e22

   1. Initial program 92.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 94.9%

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

   if 1.8499999999999999e22 < t

   1. Initial program 58.9%

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

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 82.7

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

   4. Applied rewrites 82.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 93.4

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 87.5%

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

Alternative 16: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t\_m}^{-2}\right)}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t\_m}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

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 4.5e-6)
    (/ (* (/ l k) (* l (pow t_m -2.0))) (* k t_m))
    (*
     (* 2.0 (* l l))
     (/ (/ (cos k) t_m) (* (* (- 0.5 (* 0.5 (cos (+ k k)))) k) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.5e-6) {
		tmp = ((l / k) * (l * pow(t_m, -2.0))) / (k * t_m);
	} else {
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * k) * k));
	}
	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 <= 4.5d-6) then
        tmp = ((l / k) * (l * (t_m ** (-2.0d0)))) / (k * t_m)
    else
        tmp = (2.0d0 * (l * l)) * ((cos(k) / t_m) / (((0.5d0 - (0.5d0 * cos((k + k)))) * k) * k))
    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 <= 4.5e-6) {
		tmp = ((l / k) * (l * Math.pow(t_m, -2.0))) / (k * t_m);
	} else {
		tmp = (2.0 * (l * l)) * ((Math.cos(k) / t_m) / (((0.5 - (0.5 * Math.cos((k + k)))) * k) * k));
	}
	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 <= 4.5e-6:
		tmp = ((l / k) * (l * math.pow(t_m, -2.0))) / (k * t_m)
	else:
		tmp = (2.0 * (l * l)) * ((math.cos(k) / t_m) / (((0.5 - (0.5 * math.cos((k + k)))) * k) * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.5e-6)
		tmp = Float64(Float64(Float64(l / k) * Float64(l * (t_m ^ -2.0))) / Float64(k * t_m));
	else
		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)));
	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 <= 4.5e-6)
		tmp = ((l / k) * (l * (t_m ^ -2.0))) / (k * t_m);
	else
		tmp = (2.0 * (l * l)) * ((cos(k) / t_m) / (((0.5 - (0.5 * cos((k + k)))) * k) * k));
	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, 4.5e-6], N[(N[(N[(l / k), $MachinePrecision] * N[(l * N[Power[t$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000011e-6

   1. Initial program 56.6%

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

   3. Taylor expanded in 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 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 53.9%

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

   9. Applied rewrites 66.0%

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

   if 4.50000000000000011e-6 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 62.5%

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{\cos k}{t}}{\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot k\right) \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{{\left(k \cdot k\right)}^{-1}}{t\_m} - \frac{0.16666666666666666}{t\_m}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t\_m}^{-2}\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.5e-85)
    (*
     (* 2.0 (* l l))
     (/ (- (/ (pow (* k k) -1.0) t_m) (/ 0.16666666666666666 t_m)) (* k k)))
    (/ (* (/ l k) (* l (pow t_m -2.0))) (* k t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-85) {
		tmp = (2.0 * (l * l)) * (((pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	} else {
		tmp = ((l / k) * (l * pow(t_m, -2.0))) / (k * t_m);
	}
	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 <= 2.5d-85) then
        tmp = (2.0d0 * (l * l)) * (((((k * k) ** (-1.0d0)) / t_m) - (0.16666666666666666d0 / t_m)) / (k * k))
    else
        tmp = ((l / k) * (l * (t_m ** (-2.0d0)))) / (k * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-85) {
		tmp = (2.0 * (l * l)) * (((Math.pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	} else {
		tmp = ((l / k) * (l * Math.pow(t_m, -2.0))) / (k * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.5e-85:
		tmp = (2.0 * (l * l)) * (((math.pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k))
	else:
		tmp = ((l / k) * (l * math.pow(t_m, -2.0))) / (k * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.5e-85)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(Float64(Float64((Float64(k * k) ^ -1.0) / t_m) - Float64(0.16666666666666666 / t_m)) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l * (t_m ^ -2.0))) / Float64(k * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.5e-85)
		tmp = (2.0 * (l * l)) * (((((k * k) ^ -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	else
		tmp = ((l / k) * (l * (t_m ^ -2.0))) / (k * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.5e-85], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(k * k), $MachinePrecision], -1.0], $MachinePrecision] / t$95$m), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l * N[Power[t$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5000000000000001e-85

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 51.5%

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

   if 2.5000000000000001e-85 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 75.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{{\left(k \cdot k\right)}^{-1}}{t} - \frac{0.16666666666666666}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-2}\right)}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-81}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{{\left(k \cdot k\right)}^{-1}}{t\_m} - \frac{0.16666666666666666}{t\_m}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left({t\_m}^{-3} \cdot \frac{\ell}{k}\right)}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-81)
    (*
     (* 2.0 (* l l))
     (/ (- (/ (pow (* k k) -1.0) t_m) (/ 0.16666666666666666 t_m)) (* k k)))
    (/ (* l (* (pow t_m -3.0) (/ l k))) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-81) {
		tmp = (2.0 * (l * l)) * (((pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	} else {
		tmp = (l * (pow(t_m, -3.0) * (l / k))) / k;
	}
	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 <= 3.5d-81) then
        tmp = (2.0d0 * (l * l)) * (((((k * k) ** (-1.0d0)) / t_m) - (0.16666666666666666d0 / t_m)) / (k * k))
    else
        tmp = (l * ((t_m ** (-3.0d0)) * (l / k))) / k
    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 <= 3.5e-81) {
		tmp = (2.0 * (l * l)) * (((Math.pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	} else {
		tmp = (l * (Math.pow(t_m, -3.0) * (l / k))) / k;
	}
	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 <= 3.5e-81:
		tmp = (2.0 * (l * l)) * (((math.pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k))
	else:
		tmp = (l * (math.pow(t_m, -3.0) * (l / k))) / k
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-81)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(Float64(Float64((Float64(k * k) ^ -1.0) / t_m) - Float64(0.16666666666666666 / t_m)) / Float64(k * k)));
	else
		tmp = Float64(Float64(l * Float64((t_m ^ -3.0) * Float64(l / k))) / k);
	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 <= 3.5e-81)
		tmp = (2.0 * (l * l)) * (((((k * k) ^ -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	else
		tmp = (l * ((t_m ^ -3.0) * (l / k))) / k;
	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, 3.5e-81], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(k * k), $MachinePrecision], -1.0], $MachinePrecision] / t$95$m), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[Power[t$95$m, -3.0], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.49999999999999986e-81

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 51.5%

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

   if 3.49999999999999986e-81 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   7. Applied rewrites 67.5%

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

   9. Applied rewrites 71.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-81}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{{\left(k \cdot k\right)}^{-1}}{t} - \frac{0.16666666666666666}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left({t}^{-3} \cdot \frac{\ell}{k}\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-85}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{{\left(k \cdot k\right)}^{-1}}{t\_m} - \frac{0.16666666666666666}{t\_m}}{k \cdot k}\\ \mathbf{elif}\;t\_m \leq 2.95 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot t\_m} \cdot \ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-85)
    (*
     (* 2.0 (* l l))
     (/ (- (/ (pow (* k k) -1.0) t_m) (/ 0.16666666666666666 t_m)) (* k k)))
    (if (<= t_m 2.95e+75)
      (* (/ l (* (* t_m t_m) t_m)) (/ (/ l k) k))
      (/ (/ (* (/ (/ l t_m) (* t_m t_m)) l) k) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.55e-85) {
		tmp = (2.0 * (l * l)) * (((pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	} else if (t_m <= 2.95e+75) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = ((((l / t_m) / (t_m * t_m)) * l) / k) / k;
	}
	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 <= 1.55d-85) then
        tmp = (2.0d0 * (l * l)) * (((((k * k) ** (-1.0d0)) / t_m) - (0.16666666666666666d0 / t_m)) / (k * k))
    else if (t_m <= 2.95d+75) then
        tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
    else
        tmp = ((((l / t_m) / (t_m * t_m)) * l) / k) / k
    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 <= 1.55e-85) {
		tmp = (2.0 * (l * l)) * (((Math.pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	} else if (t_m <= 2.95e+75) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = ((((l / t_m) / (t_m * t_m)) * l) / k) / k;
	}
	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 <= 1.55e-85:
		tmp = (2.0 * (l * l)) * (((math.pow((k * k), -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k))
	elif t_m <= 2.95e+75:
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
	else:
		tmp = ((((l / t_m) / (t_m * t_m)) * l) / k) / k
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.55e-85)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(Float64(Float64((Float64(k * k) ^ -1.0) / t_m) - Float64(0.16666666666666666 / t_m)) / Float64(k * k)));
	elseif (t_m <= 2.95e+75)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l / t_m) / Float64(t_m * t_m)) * l) / k) / k);
	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 <= 1.55e-85)
		tmp = (2.0 * (l * l)) * (((((k * k) ^ -1.0) / t_m) - (0.16666666666666666 / t_m)) / (k * k));
	elseif (t_m <= 2.95e+75)
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	else
		tmp = ((((l / t_m) / (t_m * t_m)) * l) / k) / k;
	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, 1.55e-85], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(k * k), $MachinePrecision], -1.0], $MachinePrecision] / t$95$m), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.95e+75], 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[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.5500000000000001e-85

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 51.5%

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

   if 1.5500000000000001e-85 < t < 2.94999999999999991e75

   1. Initial program 75.1%

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

   3. Taylor expanded in 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 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.0%

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

   9. Applied rewrites 73.9%

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

   if 2.94999999999999991e75 < t

   1. Initial program 59.0%

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

   3. Taylor expanded in 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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 67.7%

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

   9. Applied rewrites 69.9%

      \[\leadsto \frac{\frac{\frac{\frac{\ell}{t}}{t \cdot t} \cdot \ell}{k}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-85}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\frac{{\left(k \cdot k\right)}^{-1}}{t} - \frac{0.16666666666666666}{t}}{k \cdot k}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t}}{t \cdot t} \cdot \ell}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot t\_m} \cdot \ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

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 1.55)
    (/ (/ (* (/ (/ 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 <= 1.55) {
		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 <= 1.55d0) 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 <= 1.55) {
		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 <= 1.55:
		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 <= 1.55)
		tmp = Float64(Float64(Float64(Float64(Float64(l / t_m) / Float64(t_m * t_m)) * l) / k) / k);
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 / Float64(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 <= 1.55)
		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, 1.55], N[(N[(N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[(-0.16666666666666666 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004

   1. Initial program 56.6%

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

   3. Taylor expanded in 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 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 60.8%

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

   9. Applied rewrites 62.9%

      \[\leadsto \frac{\frac{\frac{\frac{\ell}{t}}{t \cdot t} \cdot \ell}{k}}{k} \]

   if 1.55000000000000004 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   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 22.8%

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

   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 51.5%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.2%

       \[\leadsto \left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 60.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

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 1.55)
    (* (/ 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 <= 1.55) {
		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 <= 1.55d0) 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 <= 1.55) {
		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 <= 1.55:
		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 <= 1.55)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 / Float64(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 <= 1.55)
		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, 1.55], 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[(-0.16666666666666666 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004

   1. Initial program 56.6%

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

   3. Taylor expanded in 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 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 53.9%

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

   9. Applied rewrites 59.3%

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

   if 1.55000000000000004 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   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 22.8%

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

   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 51.5%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.2%

       \[\leadsto \left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 59.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot t\_m} \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

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 6e+100)
    (* (/ l (* t_m t_m)) (/ (/ l (* k k)) 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 <= 6e+100) {
		tmp = (l / (t_m * t_m)) * ((l / (k * k)) / 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 <= 6d+100) then
        tmp = (l / (t_m * t_m)) * ((l / (k * k)) / 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 <= 6e+100) {
		tmp = (l / (t_m * t_m)) * ((l / (k * k)) / 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 <= 6e+100:
		tmp = (l / (t_m * t_m)) * ((l / (k * k)) / 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 <= 6e+100)
		tmp = Float64(Float64(l / Float64(t_m * t_m)) * Float64(Float64(l / Float64(k * k)) / t_m));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 / Float64(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 <= 6e+100)
		tmp = (l / (t_m * t_m)) * ((l / (k * k)) / 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, 6e+100], N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999971e100

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 57.2%

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

   if 5.99999999999999971e100 < k

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.4

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

   5. Applied rewrites 52.4%

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

   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 6.8%

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

   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 48.2%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.7%

       \[\leadsto \left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 59.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

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 6e+100)
    (* (/ 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 <= 6e+100) {
		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 <= 6d+100) 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 <= 6e+100) {
		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 <= 6e+100:
		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 <= 6e+100)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(k * k)) / Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 / Float64(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 <= 6e+100)
		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, 6e+100], 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[(-0.16666666666666666 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999971e100

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 57.2%

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

   if 5.99999999999999971e100 < k

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.4

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

   5. Applied rewrites 52.4%

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

   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 6.8%

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

   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 48.2%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 57.7%

       \[\leadsto \left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 56.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

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 1.55)
    (* (/ 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 <= 1.55) {
		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 <= 1.55d0) 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 <= 1.55) {
		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 <= 1.55:
		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 <= 1.55)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 / Float64(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 <= 1.55)
		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, 1.55], 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[(-0.16666666666666666 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004

   1. Initial program 56.6%

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

   3. Taylor expanded in 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 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 53.9%

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

   if 1.55000000000000004 < k

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   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 22.8%

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

   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 51.5%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.2%

       \[\leadsto \left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 33.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+79}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(2 \cdot \ell\right)\right) \cdot \ell\\ \end{array} \end{array} \]

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 2e+79)
    (* (* 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 <= 2e+79) {
		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 <= 2d+79) 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 <= 2e+79) {
		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 <= 2e+79:
		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 <= 2e+79)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(-0.16666666666666666 / Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 / Float64(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 <= 2e+79)
		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, 2e+79], 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[(-0.16666666666666666 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.99999999999999993e79

   1. Initial program 57.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 56.2

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

   5. Applied rewrites 56.2%

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

   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 31.6%

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

   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 30.3%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   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 33.3%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]

   if 1.99999999999999993e79 < l

   1. Initial program 33.4%

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

   3. Taylor expanded in t around 0

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

      1. 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 50.8

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

   5. Applied rewrites 50.8%

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

   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 41.7%

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

   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 15.7%

       \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 25.2%

       \[\leadsto \left(\frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t} \cdot \left(2 \cdot \ell\right)\right) \cdot \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 33.1% 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 53.4%

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

3. Taylor expanded in t around 0

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

   1. 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 55.3

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

5. Applied rewrites 55.3%

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

6. 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 33.4%

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

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 27.7%

    \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

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 30.1%

    \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]
15. Add Preprocessing

Alternative 27: 31.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot t\_m}\right) \end{array} \]

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 k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 * (l * l)) * (-0.16666666666666666 / ((k * k) * t_m)));
}

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 * k) * t_m)))
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 * k) * t_m)));
}

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 * k) * t_m)))

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 * k) * t_m))))
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 * k) * t_m)));
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 * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in t around 0

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

   1. 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 55.3

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

5. Applied rewrites 55.3%

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

6. 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 33.4%

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

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 27.7%

    \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{-0.16666666666666666}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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))))