Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.8% → 95.0%

Time: 15.1s

Alternatives: 19

Speedup: 7.8×

• 27.7% 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: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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-62)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) l)))
    (/
     2.0
     (*
      (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (* (* (sin k) 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-62) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = 2.0 / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * ((sin(k) * 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-62) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k / l) * t_m) * k) / l))
    else
        tmp = 2.0d0 / ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * ((sin(k) * 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-62) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = 2.0 / ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * ((Math.sin(k) * 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-62:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k / l) * t_m) * k) / l))
	else:
		tmp = 2.0 / ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * ((math.sin(k) * 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-62)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(Float64(sin(k) * 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-62)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	else
		tmp = 2.0 / ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * ((sin(k) * 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-62], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.79999999999999967e-62

   1. Initial program 46.3%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 81.6%

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

   if 4.79999999999999967e-62 < t

   1. Initial program 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 41.3

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

   4. Applied rewrites 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. exp-diff N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. pow2 N/A

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

      12. unpow3 N/A

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

      13. lift-*.f64 N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 93.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot t\_m\right) \cdot t\_m\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 1e-77)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) l)))
    (if (<= t_m 7.5e+125)
      (*
       (/
        2.0
        (*
         (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (/ t_m l))
         (* (* (sin k) t_m) t_m)))
       l)
      (/
       2.0
       (*
        (fma (/ k t_m) (/ k t_m) 2.0)
        (* (* (* (* (/ t_m l) (sin k)) (tan k)) (/ 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 <= 1e-77) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	} else if (t_m <= 7.5e+125) {
		tmp = (2.0 / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * (t_m / l)) * ((sin(k) * t_m) * t_m))) * l;
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (((((t_m / l) * sin(k)) * tan(k)) * (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 <= 1e-77)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / l)));
	elseif (t_m <= 7.5e+125)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(t_m / l)) * Float64(Float64(sin(k) * t_m) * t_m))) * l);
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * tan(k)) * Float64(t_m / l)) * 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, 1e-77], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+125], N[(N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.9999999999999993e-78

   1. Initial program 45.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 81.7%

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

   if 9.9999999999999993e-78 < t < 7.5000000000000006e125

   1. Initial program 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.8

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

   4. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. exp-diff N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. pow2 N/A

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

      12. unpow3 N/A

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

      13. lift-*.f64 N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   10. Applied rewrites 90.0%

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

   if 7.5000000000000006e125 < t

   1. Initial program 36.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.1

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

   6. Applied rewrites 88.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 88.1

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

   8. Applied rewrites 88.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot t\right) \cdot t\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\_m\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 3.4e-61)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (* (/ k l) t_m) k) l)))
    (/
     2.0
     (*
      (fma (/ k t_m) (/ k t_m) 2.0)
      (* (* (* (* (/ t_m l) (sin k)) (tan k)) (/ 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 <= 3.4e-61) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k / l) * t_m) * k) / l));
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (((((t_m / l) * sin(k)) * tan(k)) * (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 <= 3.4e-61)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k / l) * t_m) * k) / l)));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * tan(k)) * Float64(t_m / l)) * 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, 3.4e-61], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.3999999999999998e-61

   1. Initial program 46.3%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 81.6%

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

   if 3.3999999999999998e-61 < t

   1. Initial program 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 71.7

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

   4. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.4

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

   6. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 88.4

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

   8. Applied rewrites 88.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\left(\frac{k}{\ell} \cdot t\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)}{\ell} \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 3e-61)
    (/ 2.0 (* (/ (* (* (/ k l) k) (* (tan k) (sin k))) l) t_m))
    (/
     2.0
     (*
      (fma (/ k t_m) (/ k t_m) 2.0)
      (* (* (* (* (/ t_m l) (sin k)) (tan k)) (/ 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 <= 3e-61) {
		tmp = 2.0 / (((((k / l) * k) * (tan(k) * sin(k))) / l) * t_m);
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (((((t_m / l) * sin(k)) * tan(k)) * (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 <= 3e-61)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * k) * Float64(tan(k) * sin(k))) / l) * t_m));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * tan(k)) * Float64(t_m / l)) * 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, 3e-61], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000012e-61

   1. Initial program 46.3%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 66.6%

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

   9. Applied rewrites 74.5%

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

   if 3.00000000000000012e-61 < t

   1. Initial program 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 71.7

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

   4. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.4

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

   6. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 88.4

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

   8. Applied rewrites 88.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)}{\ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \sin k\\ t_3 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(\left(t\_3 \cdot t\_m\right) \cdot t\_3\right) \cdot 2}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot t\_m}{\ell} \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\_2\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (tan k) (sin k))) (t_3 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 8.8e-16)
      (/ 2.0 (* (* (* t_3 t_m) t_3) 2.0))
      (if (<= k 4e+147)
        (/ 2.0 (* (/ (* (/ (* k k) l) t_m) l) t_2))
        (/ 2.0 (* (* (* (/ k l) (/ k l)) t_2) 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 = tan(k) * sin(k);
	double t_3 = (t_m / l) * k;
	double tmp;
	if (k <= 8.8e-16) {
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	} else if (k <= 4e+147) {
		tmp = 2.0 / (((((k * k) / l) * t_m) / l) * t_2);
	} else {
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = tan(k) * sin(k)
    t_3 = (t_m / l) * k
    if (k <= 8.8d-16) then
        tmp = 2.0d0 / (((t_3 * t_m) * t_3) * 2.0d0)
    else if (k <= 4d+147) then
        tmp = 2.0d0 / (((((k * k) / l) * t_m) / l) * t_2)
    else
        tmp = 2.0d0 / ((((k / l) * (k / l)) * t_2) * 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 t_2 = Math.tan(k) * Math.sin(k);
	double t_3 = (t_m / l) * k;
	double tmp;
	if (k <= 8.8e-16) {
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	} else if (k <= 4e+147) {
		tmp = 2.0 / (((((k * k) / l) * t_m) / l) * t_2);
	} else {
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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):
	t_2 = math.tan(k) * math.sin(k)
	t_3 = (t_m / l) * k
	tmp = 0
	if k <= 8.8e-16:
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0)
	elif k <= 4e+147:
		tmp = 2.0 / (((((k * k) / l) * t_m) / l) * t_2)
	else:
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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(tan(k) * sin(k))
	t_3 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 8.8e-16)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_3 * t_m) * t_3) * 2.0));
	elseif (k <= 4e+147)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) / l) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(k / l)) * t_2) * 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)
	t_2 = tan(k) * sin(k);
	t_3 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 8.8e-16)
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	elseif (k <= 4e+147)
		tmp = 2.0 / (((((k * k) / l) * t_m) / l) * t_2);
	else
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 8.8e-16], N[(2.0 / N[(N[(N[(t$95$3 * t$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+147], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.80000000000000001e-16

   1. Initial program 52.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 76.4%

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

   if 8.80000000000000001e-16 < k < 3.9999999999999999e147

   1. Initial program 46.3%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 71.8%

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

   7. Applied rewrites 71.8%

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

   if 3.9999999999999999e147 < k

   1. Initial program 35.3%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 51.0%

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

   7. Applied rewrites 51.0%

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

   9. Applied rewrites 84.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\ell} \cdot t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \sin k\\ t_3 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(\left(t\_3 \cdot t\_m\right) \cdot t\_3\right) \cdot 2}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{\ell} \cdot \left(\frac{k \cdot k}{\ell} \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\_2\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (tan k) (sin k))) (t_3 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 8.8e-16)
      (/ 2.0 (* (* (* t_3 t_m) t_3) 2.0))
      (if (<= k 1.5e+125)
        (/ 2.0 (* (/ t_2 l) (* (/ (* k k) l) t_m)))
        (/ 2.0 (* (* (* (/ k l) (/ k l)) t_2) 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 = tan(k) * sin(k);
	double t_3 = (t_m / l) * k;
	double tmp;
	if (k <= 8.8e-16) {
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	} else if (k <= 1.5e+125) {
		tmp = 2.0 / ((t_2 / l) * (((k * k) / l) * t_m));
	} else {
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = tan(k) * sin(k)
    t_3 = (t_m / l) * k
    if (k <= 8.8d-16) then
        tmp = 2.0d0 / (((t_3 * t_m) * t_3) * 2.0d0)
    else if (k <= 1.5d+125) then
        tmp = 2.0d0 / ((t_2 / l) * (((k * k) / l) * t_m))
    else
        tmp = 2.0d0 / ((((k / l) * (k / l)) * t_2) * 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 t_2 = Math.tan(k) * Math.sin(k);
	double t_3 = (t_m / l) * k;
	double tmp;
	if (k <= 8.8e-16) {
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	} else if (k <= 1.5e+125) {
		tmp = 2.0 / ((t_2 / l) * (((k * k) / l) * t_m));
	} else {
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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):
	t_2 = math.tan(k) * math.sin(k)
	t_3 = (t_m / l) * k
	tmp = 0
	if k <= 8.8e-16:
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0)
	elif k <= 1.5e+125:
		tmp = 2.0 / ((t_2 / l) * (((k * k) / l) * t_m))
	else:
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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(tan(k) * sin(k))
	t_3 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 8.8e-16)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_3 * t_m) * t_3) * 2.0));
	elseif (k <= 1.5e+125)
		tmp = Float64(2.0 / Float64(Float64(t_2 / l) * Float64(Float64(Float64(k * k) / l) * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(k / l)) * t_2) * 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)
	t_2 = tan(k) * sin(k);
	t_3 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 8.8e-16)
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	elseif (k <= 1.5e+125)
		tmp = 2.0 / ((t_2 / l) * (((k * k) / l) * t_m));
	else
		tmp = 2.0 / ((((k / l) * (k / l)) * t_2) * 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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 8.8e-16], N[(2.0 / N[(N[(N[(t$95$3 * t$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+125], N[(2.0 / N[(N[(t$95$2 / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.80000000000000001e-16

   1. Initial program 52.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 76.4%

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

   if 8.80000000000000001e-16 < k < 1.50000000000000008e125

   1. Initial program 47.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.0%

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

   if 1.50000000000000008e125 < k

   1. Initial program 35.1%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.5%

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

   9. Applied rewrites 85.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \sin k}{\ell} \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_2\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 1e-15)
      (/ 2.0 (* (* (* t_2 t_m) t_2) 2.0))
      (/ 2.0 (* (* (* (/ k l) (/ k l)) (* (tan k) (sin k))) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1e-15) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / ((((k / l) * (k / l)) * (tan(k) * sin(k))) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l) * k
    if (k <= 1d-15) then
        tmp = 2.0d0 / (((t_2 * t_m) * t_2) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k / l) * (k / l)) * (tan(k) * sin(k))) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1e-15) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / ((((k / l) * (k / l)) * (Math.tan(k) * Math.sin(k))) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m / l) * k
	tmp = 0
	if k <= 1e-15:
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0)
	else:
		tmp = 2.0 / ((((k / l) * (k / l)) * (math.tan(k) * math.sin(k))) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 1e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_2) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(tan(k) * sin(k))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 1e-15)
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	else
		tmp = 2.0 / ((((k / l) * (k / l)) * (tan(k) * sin(k))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1e-15], N[(2.0 / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0000000000000001e-15

   1. Initial program 52.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 76.4%

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

   if 1.0000000000000001e-15 < k

   1. Initial program 41.1%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 62.1%

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

   7. Applied rewrites 57.6%

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

   9. Applied rewrites 73.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_2\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\_m\right) \cdot \left(\tan k \cdot \sin k\right)}{\ell \cdot \ell} \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 1e-15)
      (/ 2.0 (* (* (* t_2 t_m) t_2) 2.0))
      (/ 2.0 (* (/ (* (* k t_m) (* (tan k) (sin k))) (* l l)) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1e-15) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / ((((k * t_m) * (tan(k) * sin(k))) / (l * l)) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l) * k
    if (k <= 1d-15) then
        tmp = 2.0d0 / (((t_2 * t_m) * t_2) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k * t_m) * (tan(k) * sin(k))) / (l * l)) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1e-15) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / ((((k * t_m) * (Math.tan(k) * Math.sin(k))) / (l * l)) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m / l) * k
	tmp = 0
	if k <= 1e-15:
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0)
	else:
		tmp = 2.0 / ((((k * t_m) * (math.tan(k) * math.sin(k))) / (l * l)) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 1e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_2) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) * Float64(tan(k) * sin(k))) / Float64(l * l)) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 1e-15)
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	else
		tmp = 2.0 / ((((k * t_m) * (tan(k) * sin(k))) / (l * l)) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1e-15], N[(2.0 / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0000000000000001e-15

   1. Initial program 52.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 76.4%

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

   if 1.0000000000000001e-15 < k

   1. Initial program 41.1%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 64.9%

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

   10. Applied rewrites 66.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)}{\ell \cdot \ell} \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_2\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 1e-15)
      (/ 2.0 (* (* (* t_2 t_m) t_2) 2.0))
      (/ 2.0 (* (* (* (/ k (* l l)) k) (* (tan k) (sin k))) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1e-15) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / ((((k / (l * l)) * k) * (tan(k) * sin(k))) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l) * k
    if (k <= 1d-15) then
        tmp = 2.0d0 / (((t_2 * t_m) * t_2) * 2.0d0)
    else
        tmp = 2.0d0 / ((((k / (l * l)) * k) * (tan(k) * sin(k))) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 1e-15) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / ((((k / (l * l)) * k) * (Math.tan(k) * Math.sin(k))) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m / l) * k
	tmp = 0
	if k <= 1e-15:
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0)
	else:
		tmp = 2.0 / ((((k / (l * l)) * k) * (math.tan(k) * math.sin(k))) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 1e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_2) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / Float64(l * l)) * k) * Float64(tan(k) * sin(k))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 1e-15)
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	else
		tmp = 2.0 / ((((k / (l * l)) * k) * (tan(k) * sin(k))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1e-15], N[(2.0 / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0000000000000001e-15

   1. Initial program 52.9%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 51.2%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 76.4%

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

   if 1.0000000000000001e-15 < k

   1. Initial program 41.1%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 62.1%

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

   7. Applied rewrites 57.6%

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

   9. Applied rewrites 63.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell \cdot \ell} \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_2\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= t_m 1.45e-22)
      (/
       2.0
       (*
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (* (/ t_m l) t_m) 2.0))
         (* k k))
        (/ t_m l)))
      (/ 2.0 (* (* (* t_2 t_m) t_2) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (t_m <= 1.45e-22) {
		tmp = 2.0 / ((fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m / l) * t_m) * 2.0)) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (t_m <= 1.45e-22)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m / l) * t_m) * 2.0)) * Float64(k * k)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_2) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.45e-22], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e-22

   1. Initial program 47.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 9.9

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

   4. Applied rewrites 9.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. exp-diff N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. pow-to-exp N/A

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

      11. pow2 N/A

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

      12. unpow3 N/A

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

      13. lift-*.f64 N/A

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

      14. frac-times N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 65.1%

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

   if 1.4500000000000001e-22 < t

   1. Initial program 55.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 43.9%

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

   9. Applied rewrites 73.2%

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

   11. Applied rewrites 77.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \left(\frac{t}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{\ell}}{\ell}\\ t_3 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(t\_2 \cdot 0.16666666666666666\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_3 \cdot t\_m\right) \cdot t\_3\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ t_m l) l)) (t_3 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= t_m 1.86e-65)
      (/
       2.0
       (* (* (fma (* (* t_2 0.16666666666666666) k) k t_2) (* k k)) (* k k)))
      (/ 2.0 (* (* (* t_3 t_m) t_3) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) / l;
	double t_3 = (t_m / l) * k;
	double tmp;
	if (t_m <= 1.86e-65) {
		tmp = 2.0 / ((fma(((t_2 * 0.16666666666666666) * k), k, t_2) * (k * k)) * (k * k));
	} else {
		tmp = 2.0 / (((t_3 * t_m) * t_3) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) / l)
	t_3 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (t_m <= 1.86e-65)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(t_2 * 0.16666666666666666) * k), k, t_2) * Float64(k * k)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_3 * t_m) * t_3) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.86e-65], N[(2.0 / N[(N[(N[(N[(N[(t$95$2 * 0.16666666666666666), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$3 * t$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.86000000000000006e-65

   1. Initial program 46.3%

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

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 53.3%

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

   10. Applied rewrites 61.5%

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

   if 1.86000000000000006e-65 < t

   1. Initial program 57.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 49.8

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

   5. Applied rewrites 49.8%

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

   7. Applied rewrites 46.0%

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

   9. Applied rewrites 73.0%

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

   11. Applied rewrites 76.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\left(\frac{\frac{t}{\ell}}{\ell} \cdot 0.16666666666666666\right) \cdot k, k, \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 105000:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_2\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m\right) \cdot t\_m\right) \cdot \frac{\frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= k 105000.0)
      (/ 2.0 (* (* (* t_2 t_m) t_2) 2.0))
      (/ 2.0 (* (* (* (* (* k k) 2.0) t_m) t_m) (/ (/ t_m l) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 105000.0) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / l) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l) * k
    if (k <= 105000.0d0) then
        tmp = 2.0d0 / (((t_2 * t_m) * t_2) * 2.0d0)
    else
        tmp = 2.0d0 / (((((k * k) * 2.0d0) * t_m) * t_m) * ((t_m / l) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (k <= 105000.0) {
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / l) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m / l) * k
	tmp = 0
	if k <= 105000.0:
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0)
	else:
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / l) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (k <= 105000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_2) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) * t_m) * Float64(Float64(t_m / l) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m / l) * k;
	tmp = 0.0;
	if (k <= 105000.0)
		tmp = 2.0 / (((t_2 * t_m) * t_2) * 2.0);
	else
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) * t_m) * ((t_m / l) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 105000.0], N[(2.0 / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 105000

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 52.3

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

   5. Applied rewrites 52.3%

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

   7. Applied rewrites 50.0%

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

   9. Applied rewrites 73.4%

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

   11. Applied rewrites 74.5%

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

   if 105000 < k

   1. Initial program 42.8%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 41.9

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

   5. Applied rewrites 41.9%

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

   7. Applied rewrites 39.7%

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

   9. Applied rewrites 49.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 105000:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\right) \cdot t\right) \cdot \frac{\frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_2\right) \cdot 2} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k))) (* t_s (/ 2.0 (* (* (* t_2 t_m) t_2) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	return t_s * (2.0 / (((t_2 * t_m) * t_2) * 2.0));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    t_2 = (t_m / l) * k
    code = t_s * (2.0d0 / (((t_2 * t_m) * t_2) * 2.0d0))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	return t_s * (2.0 / (((t_2 * t_m) * t_2) * 2.0));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (t_m / l) * k
	return t_s * (2.0 / (((t_2 * t_m) * t_2) * 2.0))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_2) * 2.0)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	t_2 = (t_m / l) * k;
	tmp = t_s * (2.0 / (((t_2 * t_m) * t_2) * 2.0));
end

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.7%

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

9. Applied rewrites 67.5%

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

11. Applied rewrites 68.1%

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

12. Final simplification 68.1%

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

Alternative 14: 60.0% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.7%

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

9. Applied rewrites 56.9%

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

10. Final simplification 56.9%

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

Alternative 15: 59.9% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((((t_m / l) * t_m) / l) * t_m) * ((k * k) * 2.0d0)))
end function

Java

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

Python

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

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(t_m / l) * t_m) / l) * t_m) * Float64(Float64(k * k) * 2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((((t_m / l) * t_m) / l) * t_m) * ((k * k) * 2.0)));
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[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.7%

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

9. Applied rewrites 56.9%

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

10. Final simplification 56.9%

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

Alternative 16: 56.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m / ((l * l) / t_m)) * t_m) * ((k * k) * 2.0d0)))
end function

Java

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

Python

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

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(t_m / Float64(Float64(l * l) / t_m)) * t_m) * Float64(Float64(k * k) * 2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((t_m / ((l * l) / t_m)) * t_m) * ((k * k) * 2.0)));
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[(t$95$m / N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.7%

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

9. Applied rewrites 52.3%

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

10. Final simplification 52.3%

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

Alternative 17: 53.1% 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 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.7%

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

9. Applied rewrites 49.1%

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

Alternative 18: 52.6% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0d0)))
end function

Java

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

Python

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

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(t_m / Float64(l * l)) * Float64(t_m * t_m)) * Float64(Float64(k * k) * 2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
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[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.8%

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

8. Final simplification 47.8%

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

Alternative 19: 52.6% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((t_m * t_m) / (l * l)) * t_m) * ((k * k) * 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 47.7%

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

8. Final simplification 47.7%

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

Reproduce

herbie shell --seed 2024244 
(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))))