Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 95.6%

Time: 13.6s

Alternatives: 15

Speedup: 9.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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: 54.7% 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: 95.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\sin k \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

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 9e-83)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) l)))
    (/
     2.0
     (*
      (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (/ t_m l))
      (* (/ t_m l) (* (sin 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 <= 9e-83) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = 2.0 / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * (t_m / l)) * ((t_m / l) * (sin(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 <= 9d-83) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k / l) * t_m) * k) / l))
    else
        tmp = 2.0d0 / ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * (t_m / l)) * ((t_m / l) * (sin(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 <= 9e-83) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = 2.0 / ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * (t_m / l)) * ((t_m / l) * (Math.sin(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 <= 9e-83:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k / l) * t_m) * k) / l))
	else:
		tmp = 2.0 / ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * (t_m / l)) * ((t_m / l) * (math.sin(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 <= 9e-83)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(t_m / l)) * Float64(Float64(t_m / l) * Float64(sin(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 <= 9e-83)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	else
		tmp = 2.0 / ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * (t_m / l)) * ((t_m / l) * (sin(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, 9e-83], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999995e-83

   1. Initial program 46.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 79.1%

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

   if 8.99999999999999995e-83 < t

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 94.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m \cdot t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       2e+289)
    (/ 2.0 (* (* (/ (* t_m t_m) (* l l)) t_m) (* (* k k) 2.0)))
    (/ 2.0 (* (* (* (/ t_m (* l l)) 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 ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 2e+289) {
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * 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 ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 2d+289) then
        tmp = 2.0d0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0d0))
    else
        tmp = 2.0d0 / ((((t_m / (l * l)) * 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 ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 2e+289) {
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * 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 (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 2e+289:
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0))
	else:
		tmp = 2.0 / ((((t_m / (l * l)) * 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 (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 2e+289)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) / Float64(l * l)) * t_m) * Float64(Float64(k * k) * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(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 ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 2e+289)
		tmp = 2.0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0));
	else
		tmp = 2.0 / ((((t_m / (l * l)) * 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[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+289], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e289

   1. Initial program 79.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 69.4%

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

   if 2.0000000000000001e289 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 19.5%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.6

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

   10. Applied rewrites 68.6%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 53.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left({t\_m}^{3}, 2, \left(k \cdot t\_m\right) \cdot k\right)\right) \cdot \tan k}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \end{array} \end{array} \]

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 4.2e+88)
    (/
     2.0
     (/
      (* (* (/ (sin k) l) (fma (pow t_m 3.0) 2.0 (* (* k t_m) k))) (tan k))
      l))
    (if (<= t_m 4e+150)
      (/
       2.0
       (*
        (fma (/ k t_m) (/ k t_m) 2.0)
        (* (* (* (/ (* t_m t_m) l) (sin k)) (/ t_m l)) (tan k))))
      (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) 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 <= 4.2e+88) {
		tmp = 2.0 / ((((sin(k) / l) * fma(pow(t_m, 3.0), 2.0, ((k * t_m) * k))) * tan(k)) / l);
	} else if (t_m <= 4e+150) {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (((((t_m * t_m) / l) * sin(k)) * (t_m / l)) * tan(k)));
	} else {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * 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 <= 4.2e+88)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * fma((t_m ^ 3.0), 2.0, Float64(Float64(k * t_m) * k))) * tan(k)) / l));
	elseif (t_m <= 4e+150)
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * sin(k)) * Float64(t_m / l)) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * 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, 4.2e+88], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0 + N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+150], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e88

   1. Initial program 50.8%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 81.1%

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

   9. Applied rewrites 83.6%

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

   11. Applied rewrites 87.3%

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

   if 4.2e88 < t < 3.99999999999999992e150

   1. Initial program 63.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 93.8

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

   6. Applied rewrites 93.8%

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

   if 3.99999999999999992e150 < t

   1. Initial program 60.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.5

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

   5. Applied rewrites 60.5%

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

   7. Applied rewrites 56.0%

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

   9. Applied rewrites 89.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left({t}^{3}, 2, \left(k \cdot t\right) \cdot k\right)\right) \cdot \tan k}{\ell}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\left(\left(\frac{t \cdot t}{\ell} \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\_m\right) \cdot \frac{k}{\ell}}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

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 1.55e-18)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ 2.0 (* (/ (* (* k t_m) (/ k l)) l) (/ (pow (sin k) 2.0) (cos 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 <= 1.55e-18) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((((k * t_m) * (k / l)) / l) * (pow(sin(k), 2.0) / cos(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 <= 1.55d-18) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k * t_m) * (k / l)) / l) * ((sin(k) ** 2.0d0) / cos(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 <= 1.55e-18) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((((k * t_m) * (k / l)) / l) * (Math.pow(Math.sin(k), 2.0) / Math.cos(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 <= 1.55e-18:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = 2.0 / ((((k * t_m) * (k / l)) / l) * (math.pow(math.sin(k), 2.0) / math.cos(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 <= 1.55e-18)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) * Float64(k / l)) / l) * Float64((sin(k) ^ 2.0) / cos(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 <= 1.55e-18)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = 2.0 / ((((k * t_m) * (k / l)) / l) * ((sin(k) ^ 2.0) / cos(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, 1.55e-18], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000003e-18

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 81.6%

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

   if 1.55000000000000003e-18 < k

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 89.2%

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

Alternative 5: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \end{array} \end{array} \]

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 1.55e-18)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) 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 <= 1.55e-18) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / 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 <= 1.55d-18) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k / l) * t_m) * k) / 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 <= 1.55e-18) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k / l) * t_m) * k) / 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 <= 1.55e-18:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k / l) * t_m) * k) / 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 <= 1.55e-18)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / 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 <= 1.55e-18)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k / l) * t_m) * k) / 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, 1.55e-18], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000003e-18

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 81.6%

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

   if 1.55000000000000003e-18 < k

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 89.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({t\_2}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot t\_2\right) \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 1.55e-18)
      (/ 2.0 (* (* (pow t_2 2.0) t_m) 2.0))
      (/ 2.0 (* (* (* (/ (sin k) l) (tan k)) t_2) 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 t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1.55e-18) {
		tmp = 2.0 / ((pow(t_2, 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * t_2) * 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) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l) * k
    if (k <= 1.55d-18) then
        tmp = 2.0d0 / (((t_2 ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = 2.0d0 / ((((sin(k) / l) * tan(k)) * t_2) * 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 t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1.55e-18) {
		tmp = 2.0 / ((Math.pow(t_2, 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / ((((Math.sin(k) / l) * Math.tan(k)) * t_2) * 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):
	t_2 = (t_m / l) * k
	tmp = 0
	if k <= 1.55e-18:
		tmp = 2.0 / ((math.pow(t_2, 2.0) * t_m) * 2.0)
	else:
		tmp = 2.0 / ((((math.sin(k) / l) * math.tan(k)) * t_2) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 1.55e-18)
		tmp = Float64(2.0 / Float64(Float64((t_2 ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * t_2) * 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)
	t_2 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 1.55e-18)
		tmp = 2.0 / (((t_2 ^ 2.0) * t_m) * 2.0);
	else
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * t_2) * 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_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.55e-18], N[(2.0 / N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000003e-18

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 81.6%

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

   if 1.55000000000000003e-18 < k

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.2

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

   10. Applied rewrites 78.2%

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

   12. Applied rewrites 94.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= k 5.2e-18)
    (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))
    (/ 2.0 (* (* (* (tan k) (sin k)) (/ (/ (* k k) l) l)) 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 (k <= 5.2e-18) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / (((tan(k) * sin(k)) * (((k * k) / l) / l)) * 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 (k <= 5.2d-18) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * 2.0d0)
    else
        tmp = 2.0d0 / (((tan(k) * sin(k)) * (((k * k) / l) / l)) * 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 (k <= 5.2e-18) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	} else {
		tmp = 2.0 / (((Math.tan(k) * Math.sin(k)) * (((k * k) / l) / l)) * 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 k <= 5.2e-18:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * 2.0)
	else:
		tmp = 2.0 / (((math.tan(k) * math.sin(k)) * (((k * k) / l) / l)) * 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 (k <= 5.2e-18)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * sin(k)) * Float64(Float64(Float64(k * k) / l) / l)) * 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 (k <= 5.2e-18)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * 2.0);
	else
		tmp = 2.0 / (((tan(k) * sin(k)) * (((k * k) / l) / l)) * 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[k, 5.2e-18], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000001e-18

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 81.6%

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

   if 5.2000000000000001e-18 < k

   1. Initial program 34.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 78.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot t\_2\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ t_m l) l)))
   (*
    t_s
    (if (<= t_m 1.1e-83)
      (/
       2.0
       (* (* (fma (* (* 0.16666666666666666 t_2) k) k t_2) (* k k)) (* k k)))
      (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) / l;
	double tmp;
	if (t_m <= 1.1e-83) {
		tmp = 2.0 / ((fma(((0.16666666666666666 * t_2) * k), k, t_2) * (k * k)) * (k * k));
	} else {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) / l)
	tmp = 0.0
	if (t_m <= 1.1e-83)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(0.16666666666666666 * t_2) * k), k, t_2) * Float64(k * k)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-83], N[(2.0 / N[(N[(N[(N[(N[(0.16666666666666666 * t$95$2), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.10000000000000004e-83

   1. Initial program 46.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{{k}^{4} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{t}{{\ell}^{2}} - \frac{-1}{2} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.6%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{t}{\ell}}{\ell}\right) \cdot \color{blue}{{k}^{4}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.6%

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

   if 1.10000000000000004e-83 < t

   1. Initial program 67.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 60.8%

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

   9. Applied rewrites 82.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot t\_2\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ t_m l) l)))
   (*
    t_s
    (if (<= t_m 7.2e-60)
      (/
       2.0
       (* (* (fma (* (* 0.16666666666666666 t_2) k) k t_2) (* k k)) (* k k)))
      (/ 2.0 (* (* (* (* (* k k) 2.0) t_m) (/ t_m l)) (/ t_m 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 t_2 = (t_m / l) / l;
	double tmp;
	if (t_m <= 7.2e-60) {
		tmp = 2.0 / ((fma(((0.16666666666666666 * t_2) * k), k, t_2) * (k * k)) * (k * k));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) / l)
	tmp = 0.0
	if (t_m <= 7.2e-60)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(0.16666666666666666 * t_2) * k), k, t_2) * Float64(k * k)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * Float64(t_m / l)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-60], N[(2.0 / N[(N[(N[(N[(N[(0.16666666666666666 * t$95$2), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e-60

   1. Initial program 46.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{{k}^{4} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{t}{{\ell}^{2}} - \frac{-1}{2} \cdot \frac{t}{{\ell}^{2}}\right) + \frac{t}{{\ell}^{2}}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.2%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{t}{\ell}}{\ell}\right) \cdot \color{blue}{{k}^{4}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.1%

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

   if 7.2e-60 < t

   1. Initial program 67.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 62.3

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

   5. Applied rewrites 62.3%

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 69.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot k, k, \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell \cdot \ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(t\_2 \cdot 0.16666666666666666\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (* l l))))
   (*
    t_s
    (if (<= t_m 1.05e-61)
      (/
       2.0
       (* (* (fma (* (* t_2 0.16666666666666666) k) k t_2) (* k k)) (* k k)))
      (/ 2.0 (* (* (* (* (* k k) 2.0) t_m) (/ t_m l)) (/ t_m 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 t_2 = t_m / (l * l);
	double tmp;
	if (t_m <= 1.05e-61) {
		tmp = 2.0 / ((fma(((t_2 * 0.16666666666666666) * k), k, t_2) * (k * k)) * (k * k));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / Float64(l * l))
	tmp = 0.0
	if (t_m <= 1.05e-61)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(t_2 * 0.16666666666666666) * k), k, t_2) * Float64(k * k)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * Float64(t_m / l)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-61], N[(2.0 / N[(N[(N[(N[(N[(t$95$2 * 0.16666666666666666), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e-61

   1. Initial program 46.4%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 79.1%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 67.9

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

   10. Applied rewrites 67.9%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 58.6%

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

   if 1.05e-61 < t

   1. Initial program 68.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 61.5

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

   5. Applied rewrites 61.5%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 68.4%

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

Alternative 11: 64.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 2.9e-62)
    (/ 2.0 (* (* (* (/ t_m (* l l)) k) k) (* k k)))
    (/ 2.0 (* (* (* (* (* k k) 2.0) t_m) (/ t_m l)) (/ t_m 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 (t_m <= 2.9e-62) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / 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 (t_m <= 2.9d-62) then
        tmp = 2.0d0 / ((((t_m / (l * l)) * k) * k) * (k * k))
    else
        tmp = 2.0d0 / (((((k * k) * 2.0d0) * t_m) * (t_m / l)) * (t_m / 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 (t_m <= 2.9e-62) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / 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 t_m <= 2.9e-62:
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k))
	else:
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / 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 (t_m <= 2.9e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * Float64(t_m / l)) * Float64(t_m / 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 (t_m <= 2.9e-62)
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	else
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m / l)) * (t_m / 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[t$95$m, 2.9e-62], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999986e-62

   1. Initial program 46.4%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 79.1%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 67.9

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

   10. Applied rewrites 67.9%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 58.2%

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

   if 2.89999999999999986e-62 < t

   1. Initial program 68.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 61.5

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

   5. Applied rewrites 61.5%

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

   7. Applied rewrites 61.1%

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

   9. Applied rewrites 68.4%

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

Alternative 12: 62.6% accurate, 7.1× speedup

Math

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

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 (<= t_m 2e-26)
    (/ 2.0 (* (* (* (/ t_m (* l l)) k) k) (* k k)))
    (/ 2.0 (* (* (* (* (* k k) 2.0) t_m) t_m) (/ (/ t_m 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 (t_m <= 2e-26) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / 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 (t_m <= 2d-26) then
        tmp = 2.0d0 / ((((t_m / (l * l)) * k) * k) * (k * k))
    else
        tmp = 2.0d0 / (((((k * k) * 2.0d0) * t_m) * t_m) * ((t_m / 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 (t_m <= 2e-26) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / 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 t_m <= 2e-26:
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k))
	else:
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / 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 (t_m <= 2e-26)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * t_m) * Float64(Float64(t_m / 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 (t_m <= 2e-26)
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	else
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / 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[t$95$m, 2e-26], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e-26

   1. Initial program 47.3%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 58.1%

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

   if 2.0000000000000001e-26 < t

   1. Initial program 67.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 61.5%

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

   9. Applied rewrites 65.4%

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

Alternative 13: 62.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

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 3e-69)
    (/ 2.0 (* (* (* (/ t_m (* l l)) k) k) (* k k)))
    (/ 2.0 (* (* (* (/ t_m l) t_m) (/ t_m l)) (* (* k 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 <= 3e-69) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * 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 <= 3d-69) then
        tmp = 2.0d0 / ((((t_m / (l * l)) * k) * k) * (k * k))
    else
        tmp = 2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * 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 <= 3e-69) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * 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 <= 3e-69:
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k))
	else:
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * 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 <= 3e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(Float64(k * 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 <= 3e-69)
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	else
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * 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, 3e-69], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999989e-69

   1. Initial program 46.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 58.5%

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

   if 2.99999999999999989e-69 < t

   1. Initial program 67.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.6

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 65.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\frac{\ell}{t\_m} \cdot \ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

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 3.6e-69)
    (/ 2.0 (* (* (* (/ t_m (* l l)) k) k) (* k k)))
    (/ 2.0 (* (* (/ t_m (* (/ l t_m) l)) t_m) (* (* k 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 <= 3.6e-69) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * 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 <= 3.6d-69) then
        tmp = 2.0d0 / ((((t_m / (l * l)) * k) * k) * (k * k))
    else
        tmp = 2.0d0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * 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 <= 3.6e-69) {
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	} else {
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * 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 <= 3.6e-69:
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k))
	else:
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * 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 <= 3.6e-69)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / Float64(Float64(l / t_m) * l)) * t_m) * Float64(Float64(k * 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 <= 3.6e-69)
		tmp = 2.0 / ((((t_m / (l * l)) * k) * k) * (k * k));
	else
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * 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, 3.6e-69], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.60000000000000018e-69

   1. Initial program 46.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 58.5%

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

   if 3.60000000000000018e-69 < t

   1. Initial program 67.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.6

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

   5. Applied rewrites 60.6%

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

   7. Applied rewrites 60.2%

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

   9. Applied rewrites 63.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{t} \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.2% accurate, 9.6× speedup

Math

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

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 (* (* (* (/ t_m (* l l)) 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) {
	return t_s * (2.0 / ((((t_m / (l * l)) * k) * k) * (k * 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 / ((((t_m / (l * l)) * k) * k) * (k * 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 / ((((t_m / (l * l)) * k) * k) * (k * 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 / ((((t_m / (l * l)) * k) * k) * (k * 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(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * k) * k) * Float64(k * 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 / ((((t_m / (l * l)) * k) * k) * (k * 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[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

3. Taylor expanded in t around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

5. Applied rewrites 72.0%

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 66.3

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

10. Applied rewrites 66.3%

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

11. Taylor expanded in k around 0

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

13. Applied rewrites 57.9%

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

Reproduce

herbie shell --seed 2024268 
(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))))