Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 91.5%

Time: 13.0s

Alternatives: 21

Speedup: 7.1×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 91.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.4e-11)
    (/ 2.0 (* (/ (* (* (/ k l) t_m) k) (cos k)) (/ (pow (sin k) 2.0) l)))
    (/
     2.0
     (*
      (* t_m (* (/ t_m l) (* (/ (* (sin k) t_m) l) (tan k))))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-11) {
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * (pow(sin(k), 2.0) / l));
	} else {
		tmp = 2.0 / ((t_m * ((t_m / l) * (((sin(k) * t_m) / l) * tan(k)))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.4d-11) then
        tmp = 2.0d0 / (((((k / l) * t_m) * k) / cos(k)) * ((sin(k) ** 2.0d0) / l))
    else
        tmp = 2.0d0 / ((t_m * ((t_m / l) * (((sin(k) * t_m) / l) * tan(k)))) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.4e-11) {
		tmp = 2.0 / (((((k / l) * t_m) * k) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / l));
	} else {
		tmp = 2.0 / ((t_m * ((t_m / l) * (((Math.sin(k) * t_m) / l) * Math.tan(k)))) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.4e-11:
		tmp = 2.0 / (((((k / l) * t_m) * k) / math.cos(k)) * (math.pow(math.sin(k), 2.0) / l))
	else:
		tmp = 2.0 / ((t_m * ((t_m / l) * (((math.sin(k) * t_m) / l) * math.tan(k)))) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.4e-11)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * t_m) * k) / cos(k)) * Float64((sin(k) ^ 2.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(Float64(sin(k) * t_m) / l) * tan(k)))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.4e-11)
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * ((sin(k) ^ 2.0) / l));
	else
		tmp = 2.0 / ((t_m * ((t_m / l) * (((sin(k) * t_m) / l) * tan(k)))) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.4e-11], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.40000000000000009e-11

   1. Initial program 45.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 81.3%

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

   if 5.40000000000000009e-11 < t

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 76.2

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

   4. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 89.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.6

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

   6. Applied rewrites 89.6%

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

Alternative 2: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq \infty\right):\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot \frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
          (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))
   (*
    t_s
    (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
      (/ 2.0 (* (* (* (/ k l) t_m) k) (/ (* (tan k) (sin k)) l)))
      (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0);
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = 2.0 / ((((k / l) * t_m) * k) * ((tan(k) * sin(k)) / l));
	} else {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	}
	return t_s * tmp;
}

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 = (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0);
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= Double.POSITIVE_INFINITY)) {
		tmp = 2.0 / ((((k / l) * t_m) * k) * ((Math.tan(k) * Math.sin(k)) / l));
	} else {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0)
	tmp = 0
	if (t_2 <= 0.0) or not (t_2 <= math.inf):
		tmp = 2.0 / ((((k / l) * t_m) * k) * ((math.tan(k) * math.sin(k)) / l))
	else:
		tmp = 2.0 / (math.pow(((math.pow(t_m, 1.5) / l) * k), 2.0) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= Inf))
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * t_m) * k) * Float64(Float64(tan(k) * sin(k)) / l)));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0);
	tmp = 0.0;
	if ((t_2 <= 0.0) || ~((t_2 <= Inf)))
		tmp = 2.0 / ((((k / l) * t_m) * k) * ((tan(k) * sin(k)) / l));
	else
		tmp = 2.0 / (((((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(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] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, Infinity]], $MachinePrecision]], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 0.0 or +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 44.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 81.0%

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

   9. Applied rewrites 81.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 65.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 0 \lor \neg \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty\right):\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot t\right) \cdot k\right) \cdot \frac{\tan k \cdot \sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq \infty\right):\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
          (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))
   (*
    t_s
    (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
      (/ 2.0 (* t_m (* (/ (/ (* k k) l) l) (* (tan k) (sin k)))))
      (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0);
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = 2.0 / (t_m * ((((k * k) / l) / l) * (tan(k) * sin(k))));
	} else {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	}
	return t_s * tmp;
}

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 = (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0);
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= Double.POSITIVE_INFINITY)) {
		tmp = 2.0 / (t_m * ((((k * k) / l) / l) * (Math.tan(k) * Math.sin(k))));
	} else {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0)
	tmp = 0
	if (t_2 <= 0.0) or not (t_2 <= math.inf):
		tmp = 2.0 / (t_m * ((((k * k) / l) / l) * (math.tan(k) * math.sin(k))))
	else:
		tmp = 2.0 / (math.pow(((math.pow(t_m, 1.5) / l) * k), 2.0) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= Inf))
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(Float64(Float64(k * k) / l) / l) * Float64(tan(k) * sin(k)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0);
	tmp = 0.0;
	if ((t_2 <= 0.0) || ~((t_2 <= Inf)))
		tmp = 2.0 / (t_m * ((((k * k) / l) / l) * (tan(k) * sin(k))));
	else
		tmp = 2.0 / (((((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(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] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, Infinity]], $MachinePrecision]], N[(2.0 / N[(t$95$m * N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 0.0 or +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 44.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 68.6%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 65.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 0 \lor \neg \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty\right):\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       5e+291)
    (/ 2.0 (/ (* (* (* (* k k) 2.0) t_m) (* t_m t_m)) (* l l)))
    (/ 2.0 (* (/ (/ (* k k) l) l) (* t_m (* 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(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 5e+291) {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l));
	} else {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 5d+291) then
        tmp = 2.0d0 / (((((k * k) * 2.0d0) * t_m) * (t_m * t_m)) / (l * l))
    else
        tmp = 2.0d0 / ((((k * k) / l) / l) * (t_m * (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(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 5e+291) {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l));
	} else {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 5e+291:
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l))
	else:
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 5e+291)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * Float64(t_m * t_m)) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) / l) * Float64(t_m * 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 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 5e+291)
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l));
	else
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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[(N[Power[t$95$m, 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$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+291], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 5.0000000000000001e291

   1. Initial program 73.2%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 65.6%

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

   11. Applied rewrites 66.9%

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

   if 5.0000000000000001e291 < (/.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 20.9%

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

   3. Taylor expanded in t around 0

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

      1. 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 59.1%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{31}{360} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{173}{5040}, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{\frac{31}{360}}{\ell \cdot \ell}\right) \cdot k, k, \frac{\frac{1}{6}}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot {k}^{\color{blue}{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 44.3%

       \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 51.1%

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

Alternative 5: 92.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

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 7.5e-32)
    (/ 2.0 (* (/ (* (* (/ k l) t_m) k) (cos k)) (/ (pow (sin k) 2.0) l)))
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (* (* (/ t_m l) t_m) (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-32) {
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * (pow(sin(k), 2.0) / l));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * (((t_m / l) * t_m) * ((pow((k / t_m), 2.0) + 2.0) * tan(k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.5d-32) then
        tmp = 2.0d0 / (((((k / l) * t_m) * k) / cos(k)) * ((sin(k) ** 2.0d0) / l))
    else
        tmp = 2.0d0 / (((sin(k) * t_m) / l) * (((t_m / l) * t_m) * ((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-32) {
		tmp = 2.0 / (((((k / l) * t_m) * k) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / l));
	} else {
		tmp = 2.0 / (((Math.sin(k) * t_m) / l) * (((t_m / l) * t_m) * ((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.5e-32:
		tmp = 2.0 / (((((k / l) * t_m) * k) / math.cos(k)) * (math.pow(math.sin(k), 2.0) / l))
	else:
		tmp = 2.0 / (((math.sin(k) * t_m) / l) * (((t_m / l) * t_m) * ((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e-32)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * t_m) * k) / cos(k)) * Float64((sin(k) ^ 2.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.5e-32)
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * ((sin(k) ^ 2.0) / l));
	else
		tmp = 2.0 / (((sin(k) * t_m) / l) * (((t_m / l) * t_m) * ((((k / t_m) ^ 2.0) + 2.0) * tan(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-32], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999953e-32

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 71.9

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 81.3%

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

   9. Applied rewrites 81.8%

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

   if 7.49999999999999953e-32 < t

   1. Initial program 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 75.6

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 88.2%

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

Alternative 6: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(\frac{\tan k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 3.8e-138)
    (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
    (if (<= k 1.4e+61)
      (/
       2.0
       (*
        (fma (* t_m t_m) 2.0 (* k k))
        (* (/ (tan k) l) (* (/ (sin k) l) t_m))))
      (/ 2.0 (* (/ (* (* (/ k l) t_m) k) (cos k)) (/ (pow (sin k) 2.0) 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 <= 3.8e-138) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else if (k <= 1.4e+61) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * ((tan(k) / l) * ((sin(k) / l) * t_m)));
	} else {
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * (pow(sin(k), 2.0) / 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 <= 3.8e-138)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	elseif (k <= 1.4e+61)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(Float64(tan(k) / l) * Float64(Float64(sin(k) / l) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * t_m) * k) / cos(k)) * Float64((sin(k) ^ 2.0) / 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_] := N[(t$95$s * If[LessEqual[k, 3.8e-138], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+61], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.8000000000000002e-138

   1. Initial program 49.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]

   if 3.8000000000000002e-138 < k < 1.4000000000000001e61

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. 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 71.6%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 83.3%

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

   if 1.4000000000000001e61 < k

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 90.8%

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

   9. Applied rewrites 90.8%

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

Alternative 7: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(\frac{\tan k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\ell}{\left(\tan k \cdot \sin k\right) \cdot \left(\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k\right)}\right)}^{-1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.8e-138)
    (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
    (if (<= k 1.4e+61)
      (/
       2.0
       (*
        (fma (* t_m t_m) 2.0 (* k k))
        (* (/ (tan k) l) (* (/ (sin k) l) t_m))))
      (/
       2.0
       (pow (/ l (* (* (tan k) (sin k)) (* (* (/ k l) t_m) k))) -1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.8e-138) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else if (k <= 1.4e+61) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * ((tan(k) / l) * ((sin(k) / l) * t_m)));
	} else {
		tmp = 2.0 / pow((l / ((tan(k) * sin(k)) * (((k / l) * t_m) * k))), -1.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.8e-138)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	elseif (k <= 1.4e+61)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(Float64(tan(k) / l) * Float64(Float64(sin(k) / l) * t_m))));
	else
		tmp = Float64(2.0 / (Float64(l / Float64(Float64(tan(k) * sin(k)) * Float64(Float64(Float64(k / l) * t_m) * k))) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.8e-138], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.4e+61], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(l / N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.8000000000000002e-138

   1. Initial program 49.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]

   if 3.8000000000000002e-138 < k < 1.4000000000000001e61

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. 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 71.6%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 83.3%

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

   if 1.4000000000000001e61 < k

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 90.8%

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

   9. Applied rewrites 90.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\frac{\tan k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\ell}{\left(\tan k \cdot \sin k\right) \cdot \left(\left(\frac{k}{\ell} \cdot t\right) \cdot k\right)}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\cos k} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.044444444444444446}{\ell} \cdot k, k, \frac{-0.3333333333333333}{\ell}\right), k \cdot k, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{t\_m}}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-77)
    (/
     2.0
     (*
      (/ (* (* (/ k l) t_m) k) (cos k))
      (*
       (fma
        (fma (* (/ 0.044444444444444446 l) k) k (/ -0.3333333333333333 l))
        (* k k)
        (pow l -1.0))
       (* k k))))
    (if (<= t_m 3.2e+205)
      (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
      (/ 2.0 (* k (* k (/ 2.0 (/ (pow (/ t_m l) -2.0) t_m)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-77) {
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * (fma(fma(((0.044444444444444446 / l) * k), k, (-0.3333333333333333 / l)), (k * k), pow(l, -1.0)) * (k * k)));
	} else if (t_m <= 3.2e+205) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else {
		tmp = 2.0 / (k * (k * (2.0 / (pow((t_m / l), -2.0) / t_m))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.6e-77)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * t_m) * k) / cos(k)) * Float64(fma(fma(Float64(Float64(0.044444444444444446 / l) * k), k, Float64(-0.3333333333333333 / l)), Float64(k * k), (l ^ -1.0)) * Float64(k * k))));
	elseif (t_m <= 3.2e+205)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(2.0 / Float64((Float64(t_m / l) ^ -2.0) / t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-77], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.044444444444444446 / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(-0.3333333333333333 / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+205], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(2.0 / N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.6000000000000001e-77

   1. Initial program 44.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 82.2%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\cos k} \cdot \left({k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{2}{45} \cdot \frac{{k}^{2}}{\ell} - \frac{1}{3} \cdot \frac{1}{\ell}\right) + \frac{1}{\ell}\right)}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.6%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\cos k} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.044444444444444446}{\ell} \cdot k, k, \frac{-0.3333333333333333}{\ell}\right), k \cdot k, \frac{1}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 2.6000000000000001e-77 < t < 3.19999999999999996e205

   1. Initial program 56.2%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 50.7

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]

   if 3.19999999999999996e205 < t

   1. Initial program 69.0%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   9. Applied rewrites 62.5%

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

   11. Applied rewrites 94.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\cos k} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.044444444444444446}{\ell} \cdot k, k, \frac{-0.3333333333333333}{\ell}\right), k \cdot k, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t}{\ell}\right)}^{-2}}{t}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\cos k} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.044444444444444446}{\ell} \cdot k, k, \frac{-0.3333333333333333}{\ell}\right), k \cdot k, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 10^{+104}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, {\ell}^{-1}\right) \cdot k\right) \cdot k}{\ell \cdot \cos k} \cdot \left(t\_m \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{t\_m}}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.6e-90)
    (/
     2.0
     (*
      (/ (* (* (/ k l) t_m) k) (cos k))
      (*
       (fma
        (fma (* (/ 0.044444444444444446 l) k) k (/ -0.3333333333333333 l))
        (* k k)
        (pow l -1.0))
       (* k k))))
    (if (<= t_m 1e+104)
      (/
       2.0
       (*
        (/
         (* (* (fma -0.3333333333333333 (/ (* k k) l) (pow l -1.0)) k) k)
         (* l (cos k)))
        (* t_m (fma (* t_m t_m) 2.0 (* k k)))))
      (/ 2.0 (* k (* k (/ 2.0 (/ (pow (/ t_m l) -2.0) 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 <= 1.6e-90) {
		tmp = 2.0 / (((((k / l) * t_m) * k) / cos(k)) * (fma(fma(((0.044444444444444446 / l) * k), k, (-0.3333333333333333 / l)), (k * k), pow(l, -1.0)) * (k * k)));
	} else if (t_m <= 1e+104) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, ((k * k) / l), pow(l, -1.0)) * k) * k) / (l * cos(k))) * (t_m * fma((t_m * t_m), 2.0, (k * k))));
	} else {
		tmp = 2.0 / (k * (k * (2.0 / (pow((t_m / l), -2.0) / 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 <= 1.6e-90)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * t_m) * k) / cos(k)) * Float64(fma(fma(Float64(Float64(0.044444444444444446 / l) * k), k, Float64(-0.3333333333333333 / l)), Float64(k * k), (l ^ -1.0)) * Float64(k * k))));
	elseif (t_m <= 1e+104)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(k * k) / l), (l ^ -1.0)) * k) * k) / Float64(l * cos(k))) * Float64(t_m * fma(Float64(t_m * t_m), 2.0, Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(2.0 / Float64((Float64(t_m / l) ^ -2.0) / t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-90], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.044444444444444446 / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(-0.3333333333333333 / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+104], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(2.0 / N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.60000000000000004e-90

   1. Initial program 44.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.0

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 82.1%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\cos k} \cdot \left({k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{2}{45} \cdot \frac{{k}^{2}}{\ell} - \frac{1}{3} \cdot \frac{1}{\ell}\right) + \frac{1}{\ell}\right)}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.3%

       \[\leadsto \frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\cos k} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.044444444444444446}{\ell} \cdot k, k, \frac{-0.3333333333333333}{\ell}\right), k \cdot k, \frac{1}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   if 1.60000000000000004e-90 < t < 1e104

   1. Initial program 66.6%

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

   3. 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 71.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.2%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right) \cdot k\right) \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   if 1e104 < t

   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 k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 59.5%

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

   9. Applied rewrites 56.2%

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

   11. Applied rewrites 74.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\cos k} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.044444444444444446}{\ell} \cdot k, k, \frac{-0.3333333333333333}{\ell}\right), k \cdot k, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 10^{+104}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, {\ell}^{-1}\right) \cdot k\right) \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t}{\ell}\right)}^{-2}}{t}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot k}{\ell}\\ t_3 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(t\_m \cdot \frac{t\_2}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 10^{+104}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, t\_2, {\ell}^{-1}\right) \cdot k\right) \cdot k}{\ell \cdot \cos k} \cdot \left(t\_m \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{t\_m}}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* k k) l)) (t_3 (fma (* t_m t_m) 2.0 (* k k))))
   (*
    t_s
    (if (<= t_m 7.2e-105)
      (/ 2.0 (* t_3 (* t_m (/ t_2 l))))
      (if (<= t_m 1e+104)
        (/
         2.0
         (*
          (/
           (* (* (fma -0.3333333333333333 t_2 (pow l -1.0)) k) k)
           (* l (cos k)))
          (* t_m t_3)))
        (/ 2.0 (* k (* k (/ 2.0 (/ (pow (/ t_m l) -2.0) 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 t_2 = (k * k) / l;
	double t_3 = fma((t_m * t_m), 2.0, (k * k));
	double tmp;
	if (t_m <= 7.2e-105) {
		tmp = 2.0 / (t_3 * (t_m * (t_2 / l)));
	} else if (t_m <= 1e+104) {
		tmp = 2.0 / ((((fma(-0.3333333333333333, t_2, pow(l, -1.0)) * k) * k) / (l * cos(k))) * (t_m * t_3));
	} else {
		tmp = 2.0 / (k * (k * (2.0 / (pow((t_m / l), -2.0) / 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)
	t_2 = Float64(Float64(k * k) / l)
	t_3 = fma(Float64(t_m * t_m), 2.0, Float64(k * k))
	tmp = 0.0
	if (t_m <= 7.2e-105)
		tmp = Float64(2.0 / Float64(t_3 * Float64(t_m * Float64(t_2 / l))));
	elseif (t_m <= 1e+104)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(-0.3333333333333333, t_2, (l ^ -1.0)) * k) * k) / Float64(l * cos(k))) * Float64(t_m * t_3)));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(2.0 / Float64((Float64(t_m / l) ^ -2.0) / t_m)))));
	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[(k * k), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-105], N[(2.0 / N[(t$95$3 * N[(t$95$m * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+104], N[(2.0 / N[(N[(N[(N[(N[(-0.3333333333333333 * t$95$2 + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(2.0 / N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999929e-105

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 68.7%

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 66.2%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 63.5%

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

   if 7.19999999999999929e-105 < t < 1e104

   1. Initial program 65.2%

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

   3. 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 71.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, \frac{1}{\ell}\right) \cdot k\right) \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   if 1e104 < t

   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 k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 59.5%

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

   9. Applied rewrites 56.2%

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

   11. Applied rewrites 74.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(t \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 10^{+104}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{\ell}, {\ell}^{-1}\right) \cdot k\right) \cdot k}{\ell \cdot \cos k} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t}{\ell}\right)}^{-2}}{t}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(\frac{\tan k}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k\right) \cdot \frac{\tan k \cdot \sin k}{\ell}}\\ \end{array} \end{array} \]

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 3.8e-138)
    (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
    (if (<= k 1.5e+90)
      (/
       2.0
       (*
        (fma (* t_m t_m) 2.0 (* k k))
        (* (/ (tan k) l) (* (/ (sin k) l) t_m))))
      (/ 2.0 (* (* (* (/ k l) t_m) k) (/ (* (tan k) (sin 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 <= 3.8e-138) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else if (k <= 1.5e+90) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * ((tan(k) / l) * ((sin(k) / l) * t_m)));
	} else {
		tmp = 2.0 / ((((k / l) * t_m) * k) * ((tan(k) * sin(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 <= 3.8e-138)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	elseif (k <= 1.5e+90)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(Float64(tan(k) / l) * Float64(Float64(sin(k) / l) * t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * t_m) * k) * Float64(Float64(tan(k) * sin(k)) / 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_] := N[(t$95$s * If[LessEqual[k, 3.8e-138], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+90], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.8000000000000002e-138

   1. Initial program 49.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]

   if 3.8000000000000002e-138 < k < 1.49999999999999989e90

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.0%

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 76.1%

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

   11. Applied rewrites 84.8%

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

   if 1.49999999999999989e90 < k

   1. Initial program 46.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 89.8%

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

   9. Applied rewrites 89.9%

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

Alternative 12: 72.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{elif}\;k \leq 340000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(t\_m \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell \cdot \ell} \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t\_m}{\ell} \cdot k\right) \cdot \frac{{\sin k}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 5e-138)
    (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
    (if (<= k 340000.0)
      (/ 2.0 (* (fma (* t_m t_m) 2.0 (* k k)) (* t_m (/ (/ (* k k) l) l))))
      (if (<= k 1.1e+151)
        (/ 2.0 (* (* k k) (* (/ (* (tan k) (sin k)) (* l l)) t_m)))
        (/ 2.0 (* (* (/ (* k t_m) l) k) (/ (pow (sin k) 2.0) 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 <= 5e-138) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else if (k <= 340000.0) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * (t_m * (((k * k) / l) / l)));
	} else if (k <= 1.1e+151) {
		tmp = 2.0 / ((k * k) * (((tan(k) * sin(k)) / (l * l)) * t_m));
	} else {
		tmp = 2.0 / ((((k * t_m) / l) * k) * (pow(sin(k), 2.0) / 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 <= 5e-138)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	elseif (k <= 340000.0)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(t_m * Float64(Float64(Float64(k * k) / l) / l))));
	elseif (k <= 1.1e+151)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(tan(k) * sin(k)) / Float64(l * l)) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) / l) * k) * Float64((sin(k) ^ 2.0) / 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_] := N[(t$95$s * If[LessEqual[k, 5e-138], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 340000.0], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e+151], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 4.99999999999999989e-138

   1. Initial program 49.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]

   if 4.99999999999999989e-138 < k < 3.4e5

   1. Initial program 48.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 65.3%

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

   7. Applied rewrites 68.6%

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

   9. Applied rewrites 68.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 72.5%

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

   if 3.4e5 < k < 1.10000000000000003e151

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. 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 80.6%

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

   7. Applied rewrites 78.9%

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

   9. Applied rewrites 79.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 73.9%

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

   if 1.10000000000000003e151 < k

   1. Initial program 45.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 61.9%

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

Alternative 13: 72.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{k \cdot k}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{elif}\;k \leq 340000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(t\_m \cdot t\_2\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{\tan k \cdot \sin k}{\ell \cdot \ell} \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ (* k k) l) l)))
   (*
    t_s
    (if (<= k 5e-138)
      (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
      (if (<= k 340000.0)
        (/ 2.0 (* (fma (* t_m t_m) 2.0 (* k k)) (* t_m t_2)))
        (if (<= k 1.15e+151)
          (/ 2.0 (* (* k k) (* (/ (* (tan k) (sin k)) (* l l)) t_m)))
          (/ 2.0 (* t_2 (* t_m (* 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 t_2 = ((k * k) / l) / l;
	double tmp;
	if (k <= 5e-138) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else if (k <= 340000.0) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * (t_m * t_2));
	} else if (k <= 1.15e+151) {
		tmp = 2.0 / ((k * k) * (((tan(k) * sin(k)) / (l * l)) * t_m));
	} else {
		tmp = 2.0 / (t_2 * (t_m * (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)
	t_2 = Float64(Float64(Float64(k * k) / l) / l)
	tmp = 0.0
	if (k <= 5e-138)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	elseif (k <= 340000.0)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(t_m * t_2)));
	elseif (k <= 1.15e+151)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(tan(k) * sin(k)) / Float64(l * l)) * t_m)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(t_m * Float64(k * k))));
	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[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 5e-138], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 340000.0], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.15e+151], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 4.99999999999999989e-138

   1. Initial program 49.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}} \]

   if 4.99999999999999989e-138 < k < 3.4e5

   1. Initial program 48.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 65.3%

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

   7. Applied rewrites 68.6%

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

   9. Applied rewrites 68.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 72.5%

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

   if 3.4e5 < k < 1.15e151

   1. Initial program 47.1%

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

   3. Taylor expanded in t around 0

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

      1. 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 80.6%

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

   7. Applied rewrites 78.9%

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

   9. Applied rewrites 79.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 73.9%

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

   if 1.15e151 < k

   1. Initial program 45.9%

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

   3. Taylor expanded in t around 0

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

      1. 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 60.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{31}{360} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{173}{5040}, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{\frac{31}{360}}{\ell \cdot \ell}\right) \cdot k, k, \frac{\frac{1}{6}}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot {k}^{\color{blue}{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.7%

       \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 60.7%

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

Alternative 14: 70.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(t\_m \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{2}{\frac{{\left(\frac{t\_m}{\ell}\right)}^{-2}}{t\_m}}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e+67)
    (/ 2.0 (* (fma (* t_m t_m) 2.0 (* k k)) (* t_m (/ (/ (* k k) l) l))))
    (/ 2.0 (* k (* k (/ 2.0 (/ (pow (/ t_m l) -2.0) t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e+67) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * (t_m * (((k * k) / l) / l)));
	} else {
		tmp = 2.0 / (k * (k * (2.0 / (pow((t_m / l), -2.0) / t_m))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e+67)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(t_m * Float64(Float64(Float64(k * k) / l) / l))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(2.0 / Float64((Float64(t_m / l) ^ -2.0) / t_m)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e+67], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(2.0 / N[(N[Power[N[(t$95$m / l), $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.7999999999999998e67

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 68.6%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 65.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 62.2%

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

   if 2.7999999999999998e67 < t

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 64.8%

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

   9. Applied rewrites 62.0%

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

   11. Applied rewrites 78.2%

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

Alternative 15: 67.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right) \cdot \left(t\_m \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 1.85e-62)
    (/ 2.0 (* (fma (* t_m t_m) 2.0 (* k k)) (* t_m (/ (/ (* k k) l) l))))
    (/ 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 <= 1.85e-62) {
		tmp = 2.0 / (fma((t_m * t_m), 2.0, (k * k)) * (t_m * (((k * k) / l) / l)));
	} 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 <= 1.85e-62)
		tmp = Float64(2.0 / Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) * Float64(t_m * Float64(Float64(Float64(k * k) / l) / l))));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-62], N[(2.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $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 < 1.8499999999999999e-62

   1. Initial program 44.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   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 68.8%

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

   7. Applied rewrites 66.0%

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

   9. Applied rewrites 65.9%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 63.4%

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

   if 1.8499999999999999e-62 < t

   1. Initial program 60.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 58.7%

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

   9. Applied rewrites 57.1%

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

   11. Applied rewrites 62.3%

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

Alternative 16: 63.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(t\_m \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 (<= l 1.25e+31)
    (/ 2.0 (* (/ (/ (* k k) l) l) (* t_m (fma (* t_m t_m) 2.0 (* 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 (l <= 1.25e+31) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * fma((t_m * t_m), 2.0, (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 (l <= 1.25e+31)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) / l) * Float64(t_m * fma(Float64(t_m * t_m), 2.0, 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_] := N[(t$95$s * If[LessEqual[l, 1.25e+31], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $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 l < 1.25000000000000007e31

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 67.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 59.2%

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

   if 1.25000000000000007e31 < l

   1. Initial program 48.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 58.9

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

   5. Applied rewrites 58.9%

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 54.7%

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

   11. Applied rewrites 67.0%

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

Alternative 17: 66.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 4.8e-77)
    (/ 2.0 (* (/ (/ (* k k) l) l) (* t_m (* 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 <= 4.8e-77) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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 <= 4.8d-77) then
        tmp = 2.0d0 / ((((k * k) / l) / l) * (t_m * (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 <= 4.8e-77) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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 <= 4.8e-77:
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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 <= 4.8e-77)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) / l) * Float64(t_m * 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 <= 4.8e-77)
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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, 4.8e-77], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $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 < 4.7999999999999998e-77

   1. Initial program 44.3%

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

   3. Taylor expanded in t around 0

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

      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 69.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{31}{360} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.3%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{173}{5040}, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{\frac{31}{360}}{\ell \cdot \ell}\right) \cdot k, k, \frac{\frac{1}{6}}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot {k}^{\color{blue}{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.4%

       \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 55.9%

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

   if 4.7999999999999998e-77 < t

   1. Initial program 59.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 56.0%

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

   11. Applied rewrites 62.2%

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

Alternative 18: 64.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

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 2.9e-75)
    (/ 2.0 (* (/ (/ (* k k) l) l) (* t_m (* k k))))
    (/ 2.0 (* (* (* k k) 2.0) (* (/ t_m l) (* (/ t_m 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 (t_m <= 2.9e-75) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (k * k)));
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m / l) * ((t_m / 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 (t_m <= 2.9d-75) then
        tmp = 2.0d0 / ((((k * k) / l) / l) * (t_m * (k * k)))
    else
        tmp = 2.0d0 / (((k * k) * 2.0d0) * ((t_m / l) * ((t_m / 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 (t_m <= 2.9e-75) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (k * k)));
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m / l) * ((t_m / 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 t_m <= 2.9e-75:
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (k * k)))
	else:
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m / l) * ((t_m / 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 (t_m <= 2.9e-75)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) / l) * Float64(t_m * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * 2.0) * Float64(Float64(t_m / l) * Float64(Float64(t_m / 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 (t_m <= 2.9e-75)
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (k * k)));
	else
		tmp = 2.0 / (((k * k) * 2.0) * ((t_m / l) * ((t_m / 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[t$95$m, 2.9e-75], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.9000000000000002e-75

   1. Initial program 44.1%

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

   3. Taylor expanded in t around 0

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

      1. 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 68.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{31}{360} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{173}{5040}, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{\frac{31}{360}}{\ell \cdot \ell}\right) \cdot k, k, \frac{\frac{1}{6}}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot {k}^{\color{blue}{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.1%

       \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 55.5%

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

   if 2.9000000000000002e-75 < t

   1. Initial program 60.1%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 61.4%

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

Alternative 19: 64.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(t\_m \cdot \frac{\frac{t\_m}{\ell} \cdot t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e-75)
    (/ 2.0 (* (/ (/ (* k k) l) l) (* t_m (* 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-75) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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-75) then
        tmp = 2.0d0 / ((((k * k) / l) / l) * (t_m * (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-75) {
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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-75:
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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-75)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) / l) * Float64(t_m * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * 2.0) * Float64(t_m * Float64(Float64(Float64(t_m / l) * 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-75)
		tmp = 2.0 / ((((k * k) / l) / l) * (t_m * (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-75], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.9000000000000002e-75

   1. Initial program 44.1%

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

   3. Taylor expanded in t around 0

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

      1. 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 68.5%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{31}{360} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{6} \cdot \frac{1}{{\ell}^{2}}\right) + \frac{1}{{\ell}^{2}}\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{t} \cdot \mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{173}{5040}, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{\frac{31}{360}}{\ell \cdot \ell}\right) \cdot k, k, \frac{\frac{1}{6}}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot {k}^{\color{blue}{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.1%

       \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{\frac{k \cdot k}{\ell}}{\ell}, \frac{0.08611111111111111}{\ell \cdot \ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell \cdot \ell}\right), k \cdot k, \frac{\frac{1}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(k \cdot \color{blue}{k}\right)\right)} \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 55.5%

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

   if 2.9000000000000002e-75 < t

   1. Initial program 60.1%

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

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 61.4%

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

Alternative 20: 54.2% accurate, 8.7× speedup

Math

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

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 (/ (* (* (* (* 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) {
	return t_s * (2.0 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l)));
}

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 / (((((k * k) * 2.0d0) * t_m) * (t_m * t_m)) / (l * l)))
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 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l)));
}

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 / (((((k * k) * 2.0) * t_m) * (t_m * t_m)) / (l * l)))

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

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 52.5

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

5. Applied rewrites 52.5%

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

7. Applied rewrites 56.0%

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

9. Applied rewrites 51.5%

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

11. Applied rewrites 52.0%

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

Alternative 21: 53.7% accurate, 8.7× speedup

Math

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

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 (* (* (* 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) {
	return t_s * (2.0 / (((k * k) * 2.0) * (t_m * ((t_m * t_m) / (l * l)))));
}

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 / (((k * k) * 2.0d0) * (t_m * ((t_m * t_m) / (l * l)))))
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 / (((k * k) * 2.0) * (t_m * ((t_m * t_m) / (l * l)))));
}

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 / (((k * k) * 2.0) * (t_m * ((t_m * t_m) / (l * l)))))

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

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 52.5

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

5. Applied rewrites 52.5%

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

7. Applied rewrites 56.0%

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

9. Applied rewrites 51.5%

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

Reproduce

herbie shell --seed 2024307 
(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))))