Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 95.3%

Time: 13.5s

Alternatives: 21

Speedup: 10.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m \cdot k}{\ell}\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-50)
    (/ 2.0 (* (* k (/ (* t_m k) l)) (/ (/ (pow (sin k) 2.0) l) (cos k))))
    (/
     2.0
     (*
      (* (* (sin k) t_m) (/ t_m l))
      (* (/ t_m l) (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-50) {
		tmp = 2.0 / ((k * ((t_m * k) / l)) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * ((t_m / l) * (tan(k) * (pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.5d-50) then
        tmp = 2.0d0 / ((k * ((t_m * k) / l)) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    else
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * ((t_m / l) * (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-50) {
		tmp = 2.0 / ((k * ((t_m * k) / l)) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * ((t_m / l) * (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.5e-50:
		tmp = 2.0 / ((k * ((t_m * k) / l)) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	else:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * ((t_m / l) * (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-50)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(t_m * k) / l)) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(t_m / l) * Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.5e-50)
		tmp = 2.0 / ((k * ((t_m * k) / l)) * (((sin(k) ^ 2.0) / l) / cos(k)));
	else
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * ((t_m / l) * (tan(k) * (((k / t_m) ^ 2.0) + 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-50], N[(2.0 / N[(N[(k * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.49999999999999997e-50

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 83.4%

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

   if 3.49999999999999997e-50 < t

   1. Initial program 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.6%

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

Alternative 2: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{t\_2 \cdot t\_2}}\\ \end{array} \end{array} \end{array} \]

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 (/ l (* k k))))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
           (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
         2e+303)
      (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
      (/ 2.0 (/ t_m (* t_2 t_2)))))))

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 = l / (k * k);
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+303) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (t_m / (t_2 * t_2));
	}
	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 = l / (k * k)
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+303) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
    else
        tmp = 2.0d0 / (t_m / (t_2 * t_2))
    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 = l / (k * k);
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+303) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (t_m / (t_2 * t_2));
	}
	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 = l / (k * k)
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+303:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = 2.0 / (t_m / (t_2 * t_2))
	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(l / Float64(k * k))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+303)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(t_m / Float64(t_2 * t_2)));
	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 = l / (k * k);
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+303)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = 2.0 / (t_m / (t_2 * t_2));
	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[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 80.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 73.6%

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

   11. Applied rewrites 80.3%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 24.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 58.6%

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

   10. Applied rewrites 58.9%

       \[\leadsto \frac{2}{\frac{t \cdot 1}{\ell \cdot \color{blue}{\frac{\ell}{{k}^{4}}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 58.8%

       \[\leadsto \frac{2}{\frac{t \cdot 1}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot k}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{k} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+303)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
    (/ 2.0 (* (* (* k k) (/ (* k k) 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 ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+303) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (((k * k) * ((k * k) / 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 ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+303) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
    else
        tmp = 2.0d0 / (((k * k) * ((k * k) / 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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+303) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = 2.0 / (((k * k) * ((k * k) / 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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+303:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = 2.0 / (((k * k) * ((k * k) / 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 (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+303)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+303)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = 2.0 / (((k * k) * ((k * k) / 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[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e303

   1. Initial program 80.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 73.6%

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

   11. Applied rewrites 80.3%

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

   if 2e303 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 24.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 58.6%

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

   10. Applied rewrites 58.6%

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

Alternative 4: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m \cdot k}{\ell}\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-32)
    (/ 2.0 (* (* k (/ (* t_m k) l)) (/ (/ (pow (sin k) 2.0) l) (cos k))))
    (/
     2.0
     (*
      (* (fma (/ k t_m) (/ k t_m) 2.0) (/ t_m l))
      (* (* (* (sin k) t_m) (/ t_m l)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-32) {
		tmp = 2.0 / ((k * ((t_m * k) / l)) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = 2.0 / ((fma((k / t_m), (k / t_m), 2.0) * (t_m / l)) * (((sin(k) * t_m) * (t_m / l)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-32)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(Float64(t_m * k) / l)) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(t_m / l)) * Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-32], N[(2.0 / N[(N[(k * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e-32

   1. Initial program 51.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 83.2%

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

   if 1.5e-32 < t

   1. Initial program 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 78.4

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 98.7

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

   8. Applied rewrites 98.7%

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

Alternative 5: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell} \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(t\_m \cdot \mathsf{fma}\left(\frac{0.26666666666666666 + \frac{0.3333333333333333}{t\_m \cdot t\_m}}{\ell}, k \cdot k, \frac{0.6666666666666666 + {\left(t\_m \cdot t\_m\right)}^{-1}}{\ell}\right), k \cdot k, \frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.6e-48)
    (/ 2.0 (* (/ (* (* k k) t_m) l) (/ (/ (* k k) l) (cos k))))
    (/
     2.0
     (*
      (* (* (sin k) t_m) (/ t_m l))
      (*
       (fma
        (*
         t_m
         (fma
          (/ (+ 0.26666666666666666 (/ 0.3333333333333333 (* t_m t_m))) l)
          (* k k)
          (/ (+ 0.6666666666666666 (pow (* t_m t_m) -1.0)) l)))
        (* k k)
        (* (/ t_m l) 2.0))
       k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.6e-48) {
		tmp = 2.0 / ((((k * k) * t_m) / l) * (((k * k) / l) / cos(k)));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (fma((t_m * fma(((0.26666666666666666 + (0.3333333333333333 / (t_m * t_m))) / l), (k * k), ((0.6666666666666666 + pow((t_m * t_m), -1.0)) / l))), (k * k), ((t_m / l) * 2.0)) * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.6e-48)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / l) * Float64(Float64(Float64(k * k) / l) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(fma(Float64(t_m * fma(Float64(Float64(0.26666666666666666 + Float64(0.3333333333333333 / Float64(t_m * t_m))) / l), Float64(k * k), Float64(Float64(0.6666666666666666 + (Float64(t_m * t_m) ^ -1.0)) / l))), Float64(k * k), Float64(Float64(t_m / l) * 2.0)) * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.6e-48], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * N[(N[(N[(0.26666666666666666 + N[(0.3333333333333333 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(0.6666666666666666 + N[Power[N[(t$95$m * t$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.6000000000000001e-48

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 72.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 63.6%

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

   if 5.6000000000000001e-48 < t

   1. Initial program 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(k \cdot \left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right) \cdot k\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{t}{\ell} + {k}^{2} \cdot \left(\frac{t \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)}{\ell} + \frac{{k}^{2} \cdot \left(t \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right) \cdot k\right)}} \]

   9. Applied rewrites 87.7%

      \[\leadsto \frac{2}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(\frac{0.26666666666666666 + \frac{0.3333333333333333}{t \cdot t}}{\ell}, k \cdot k, \frac{0.6666666666666666 + \frac{1}{t \cdot t}}{\ell}\right), k \cdot k, \frac{t}{\ell} \cdot 2\right) \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(\frac{0.26666666666666666 + \frac{0.3333333333333333}{t \cdot t}}{\ell}, k \cdot k, \frac{0.6666666666666666 + {\left(t \cdot t\right)}^{-1}}{\ell}\right), k \cdot k, \frac{t}{\ell} \cdot 2\right) \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-24)
    (/ 2.0 (/ (* (* (/ (sin k) l) (* (tan k) k)) (* t_m k)) l))
    (/
     2.0
     (*
      (* (fma (/ k t_m) (/ k t_m) 2.0) (/ t_m l))
      (* (* (* (sin k) t_m) (/ t_m l)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-24) {
		tmp = 2.0 / ((((sin(k) / l) * (tan(k) * k)) * (t_m * k)) / l);
	} else {
		tmp = 2.0 / ((fma((k / t_m), (k / t_m), 2.0) * (t_m / l)) * (((sin(k) * t_m) * (t_m / l)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e-24)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * Float64(tan(k) * k)) * Float64(t_m * k)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(t_m / l)) * Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-24], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.24999999999999995e-24

   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 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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.6

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 72.3%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 82.4%

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

   if 1.24999999999999995e-24 < t

   1. Initial program 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 77.8

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

   4. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

Alternative 7: 93.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{\ell \cdot t\_m}, \frac{2}{\ell} \cdot t\_m\right) \cdot \left(\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.8e-51)
    (/ 2.0 (/ (* (* (/ (sin k) l) (* (tan k) k)) (* t_m k)) l))
    (/
     2.0
     (*
      (fma k (/ k (* l t_m)) (* (/ 2.0 l) t_m))
      (* (* (* (sin k) t_m) (/ t_m l)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.8e-51) {
		tmp = 2.0 / ((((sin(k) / l) * (tan(k) * k)) * (t_m * k)) / l);
	} else {
		tmp = 2.0 / (fma(k, (k / (l * t_m)), ((2.0 / l) * t_m)) * (((sin(k) * t_m) * (t_m / l)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.8e-51)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * Float64(tan(k) * k)) * Float64(t_m * k)) / l));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(l * t_m)), Float64(Float64(2.0 / l) * t_m)) * Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.8e-51], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(l * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.79999999999999948e-51

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 72.0%

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

   11. Applied rewrites 82.4%

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

   if 9.79999999999999948e-51 < t

   1. Initial program 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 93.1

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

   9. Applied rewrites 93.1%

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

Alternative 8: 78.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.1e-52)
    (/ 2.0 (* (* (* (sin k) t_m) (/ t_m l)) (* (* (/ t_m l) 2.0) k)))
    (/ 2.0 (/ (* (* (/ (sin k) l) (* (tan k) k)) (* t_m k)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.1e-52) {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / ((((sin(k) / l) * (tan(k) * k)) * (t_m * k)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d-52) then
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0d0) * k))
    else
        tmp = 2.0d0 / ((((sin(k) / l) * (tan(k) * k)) * (t_m * k)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.1e-52) {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / ((((Math.sin(k) / l) * (Math.tan(k) * k)) * (t_m * k)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.1e-52:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k))
	else:
		tmp = 2.0 / ((((math.sin(k) / l) * (math.tan(k) * k)) * (t_m * k)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.1e-52)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(Float64(t_m / l) * 2.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * Float64(tan(k) * k)) * Float64(t_m * k)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.1e-52)
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	else
		tmp = 2.0 / ((((sin(k) / l) * (tan(k) * k)) * (t_m * k)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.1e-52], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.0999999999999999e-52

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if 3.0999999999999999e-52 < k

   1. Initial program 49.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 79.4%

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

   11. Applied rewrites 89.9%

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

Alternative 9: 77.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot k\right)\right) \cdot \left(k \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.6e-53)
    (/ 2.0 (* (* (* (sin k) t_m) (/ t_m l)) (* (* (/ t_m l) 2.0) k)))
    (/ 2.0 (* (* (/ (sin k) l) (* (tan k) k)) (* k (/ 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 (k <= 2.6e-53) {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / (((sin(k) / l) * (tan(k) * k)) * (k * (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 (k <= 2.6d-53) then
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0d0) * k))
    else
        tmp = 2.0d0 / (((sin(k) / l) * (tan(k) * k)) * (k * (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 (k <= 2.6e-53) {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / (((Math.sin(k) / l) * (Math.tan(k) * k)) * (k * (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 k <= 2.6e-53:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k))
	else:
		tmp = 2.0 / (((math.sin(k) / l) * (math.tan(k) * k)) * (k * (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 (k <= 2.6e-53)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(Float64(t_m / l) * 2.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * Float64(tan(k) * k)) * Float64(k * 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 (k <= 2.6e-53)
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	else
		tmp = 2.0 / (((sin(k) / l) * (tan(k) * k)) * (k * (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[k, 2.6e-53], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.59999999999999996e-53

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if 2.59999999999999996e-53 < k

   1. Initial program 49.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 79.4%

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

   11. Applied rewrites 89.3%

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

Alternative 10: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\frac{\sin k}{\ell} \cdot \left(\tan k \cdot k\right)\right)\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.6e-53)
    (/ 2.0 (* (* (* (sin k) t_m) (/ t_m l)) (* (* (/ t_m l) 2.0) k)))
    (/ 2.0 (* (* (/ t_m l) (* (/ (sin k) l) (* (tan k) k))) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-53) {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / (((t_m / l) * ((sin(k) / l) * (tan(k) * k))) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.6d-53) then
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0d0) * k))
    else
        tmp = 2.0d0 / (((t_m / l) * ((sin(k) / l) * (tan(k) * k))) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-53) {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / (((t_m / l) * ((Math.sin(k) / l) * (Math.tan(k) * k))) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.6e-53:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k))
	else:
		tmp = 2.0 / (((t_m / l) * ((math.sin(k) / l) * (math.tan(k) * k))) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.6e-53)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(Float64(t_m / l) * 2.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(sin(k) / l) * Float64(tan(k) * k))) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.6e-53)
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	else
		tmp = 2.0 / (((t_m / l) * ((sin(k) / l) * (tan(k) * k))) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.6e-53], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.59999999999999996e-53

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if 2.59999999999999996e-53 < k

   1. Initial program 49.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 79.4%

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

   11. Applied rewrites 89.3%

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

Alternative 11: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

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 (<= k 8.5e-53)
    (/ 2.0 (* (* (* (sin k) t_m) (/ t_m l)) (* (* (/ t_m l) 2.0) k)))
    (/ 2.0 (* (* (* (/ (sin k) l) (tan k)) (* k k)) (/ 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 (k <= 8.5e-53) {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * (k * k)) * (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 (k <= 8.5d-53) then
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0d0) * k))
    else
        tmp = 2.0d0 / ((((sin(k) / l) * tan(k)) * (k * k)) * (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 (k <= 8.5e-53) {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / ((((Math.sin(k) / l) * Math.tan(k)) * (k * k)) * (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 k <= 8.5e-53:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k))
	else:
		tmp = 2.0 / ((((math.sin(k) / l) * math.tan(k)) * (k * k)) * (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 (k <= 8.5e-53)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(Float64(t_m / l) * 2.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(k * k)) * 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 (k <= 8.5e-53)
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	else
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * (k * k)) * (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[k, 8.5e-53], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.50000000000000044e-53

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if 8.50000000000000044e-53 < k

   1. Initial program 49.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 79.4%

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

Alternative 12: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot t\_m}{\ell} \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.1e-52)
    (/ 2.0 (* (* (* (sin k) t_m) (/ t_m l)) (* (* (/ t_m l) 2.0) k)))
    (/ 2.0 (* (/ (* (* k k) t_m) l) (/ (/ (* k k) l) (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.1e-52) {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / l) * (((k * k) / l) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d-52) then
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0d0) * k))
    else
        tmp = 2.0d0 / ((((k * k) * t_m) / l) * (((k * k) / l) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.1e-52) {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	} else {
		tmp = 2.0 / ((((k * k) * t_m) / l) * (((k * k) / l) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.1e-52:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k))
	else:
		tmp = 2.0 / ((((k * k) * t_m) / l) * (((k * k) / l) / math.cos(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.1e-52)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(Float64(t_m / l) * 2.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) / l) * Float64(Float64(Float64(k * k) / l) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.1e-52)
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	else
		tmp = 2.0 / ((((k * k) * t_m) / l) * (((k * k) / l) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.1e-52], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.0999999999999999e-52

   1. Initial program 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 71.2

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

   4. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if 3.0999999999999999e-52 < k

   1. Initial program 49.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 73.5

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 67.1%

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

Alternative 13: 75.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-51)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (/ 2.0 (* (* (* (sin k) t_m) (/ t_m l)) (* (* (/ t_m l) 2.0) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-51) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4d-51) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else
        tmp = 2.0d0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0d0) * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-51) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / (((Math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4e-51:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	else:
		tmp = 2.0 / (((math.sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e-51)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) * Float64(t_m / l)) * Float64(Float64(Float64(t_m / l) * 2.0) * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4e-51)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	else
		tmp = 2.0 / (((sin(k) * t_m) * (t_m / l)) * (((t_m / l) * 2.0) * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-51], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4e-51

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 62.7%

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

   10. Applied rewrites 63.8%

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

   if 4e-51 < t

   1. Initial program 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 79.4

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 85.6

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

   9. Applied rewrites 85.6%

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

Alternative 14: 73.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{\ell}{{t\_m}^{3}} \cdot \frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t\_m \cdot k\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-95)
    (/ 2.0 (* (* (* k k) t_m) (/ (/ (* k k) l) (* (cos k) l))))
    (if (<= t_m 1.6e+71)
      (/ (* (/ l (pow t_m 3.0)) (/ l k)) k)
      (* (/ l (pow (* t_m k) 2.0)) (/ 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 <= 4.2e-95) {
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * l)));
	} else if (t_m <= 1.6e+71) {
		tmp = ((l / pow(t_m, 3.0)) * (l / k)) / k;
	} else {
		tmp = (l / pow((t_m * k), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-95) then
        tmp = 2.0d0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * l)))
    else if (t_m <= 1.6d+71) then
        tmp = ((l / (t_m ** 3.0d0)) * (l / k)) / k
    else
        tmp = (l / ((t_m * k) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-95) {
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (Math.cos(k) * l)));
	} else if (t_m <= 1.6e+71) {
		tmp = ((l / Math.pow(t_m, 3.0)) * (l / k)) / k;
	} else {
		tmp = (l / Math.pow((t_m * k), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-95:
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (math.cos(k) * l)))
	elif t_m <= 1.6e+71:
		tmp = ((l / math.pow(t_m, 3.0)) * (l / k)) / k
	else:
		tmp = (l / math.pow((t_m * k), 2.0)) * (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 <= 4.2e-95)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(Float64(k * k) / l) / Float64(cos(k) * l))));
	elseif (t_m <= 1.6e+71)
		tmp = Float64(Float64(Float64(l / (t_m ^ 3.0)) * Float64(l / k)) / k);
	else
		tmp = Float64(Float64(l / (Float64(t_m * k) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-95)
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * l)));
	elseif (t_m <= 1.6e+71)
		tmp = ((l / (t_m ^ 3.0)) * (l / k)) / k;
	else
		tmp = (l / ((t_m * k) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-95], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+71], N[(N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(l / N[Power[N[(t$95$m * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e-95

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 62.3%

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

   if 4.2e-95 < t < 1.60000000000000012e71

   1. Initial program 78.5%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 81.6%

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

   if 1.60000000000000012e71 < t

   1. Initial program 55.6%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 60.0%

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

   11. Applied rewrites 88.2%

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

Alternative 15: 74.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k \cdot k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{t\_2 \cdot t\_2}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{\ell}{{t\_m}^{3}} \cdot \frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t\_m \cdot k\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ l (* k k))))
   (*
    t_s
    (if (<= t_m 4e-95)
      (/ 2.0 (/ t_m (* t_2 t_2)))
      (if (<= t_m 1.6e+71)
        (/ (* (/ l (pow t_m 3.0)) (/ l k)) k)
        (* (/ l (pow (* t_m k) 2.0)) (/ 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 t_2 = l / (k * k);
	double tmp;
	if (t_m <= 4e-95) {
		tmp = 2.0 / (t_m / (t_2 * t_2));
	} else if (t_m <= 1.6e+71) {
		tmp = ((l / pow(t_m, 3.0)) * (l / k)) / k;
	} else {
		tmp = (l / pow((t_m * k), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (k * k)
    if (t_m <= 4d-95) then
        tmp = 2.0d0 / (t_m / (t_2 * t_2))
    else if (t_m <= 1.6d+71) then
        tmp = ((l / (t_m ** 3.0d0)) * (l / k)) / k
    else
        tmp = (l / ((t_m * k) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (k * k);
	double tmp;
	if (t_m <= 4e-95) {
		tmp = 2.0 / (t_m / (t_2 * t_2));
	} else if (t_m <= 1.6e+71) {
		tmp = ((l / Math.pow(t_m, 3.0)) * (l / k)) / k;
	} else {
		tmp = (l / Math.pow((t_m * k), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l / (k * k)
	tmp = 0
	if t_m <= 4e-95:
		tmp = 2.0 / (t_m / (t_2 * t_2))
	elif t_m <= 1.6e+71:
		tmp = ((l / math.pow(t_m, 3.0)) * (l / k)) / k
	else:
		tmp = (l / math.pow((t_m * k), 2.0)) * (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)
	t_2 = Float64(l / Float64(k * k))
	tmp = 0.0
	if (t_m <= 4e-95)
		tmp = Float64(2.0 / Float64(t_m / Float64(t_2 * t_2)));
	elseif (t_m <= 1.6e+71)
		tmp = Float64(Float64(Float64(l / (t_m ^ 3.0)) * Float64(l / k)) / k);
	else
		tmp = Float64(Float64(l / (Float64(t_m * k) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l / (k * k);
	tmp = 0.0;
	if (t_m <= 4e-95)
		tmp = 2.0 / (t_m / (t_2 * t_2));
	elseif (t_m <= 1.6e+71)
		tmp = ((l / (t_m ^ 3.0)) * (l / k)) / k;
	else
		tmp = (l / ((t_m * k) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-95], N[(2.0 / N[(t$95$m / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+71], N[(N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(l / N[Power[N[(t$95$m * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.99999999999999996e-95

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 62.5%

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

   10. Applied rewrites 62.2%

       \[\leadsto \frac{2}{\frac{t \cdot 1}{\ell \cdot \color{blue}{\frac{\ell}{{k}^{4}}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.2%

       \[\leadsto \frac{2}{\frac{t \cdot 1}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot k}}}} \]

   if 3.99999999999999996e-95 < t < 1.60000000000000012e71

   1. Initial program 78.5%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 81.6%

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

   if 1.60000000000000012e71 < t

   1. Initial program 55.6%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

   9. Applied rewrites 60.0%

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

   11. Applied rewrites 88.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot k\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(t\_m \cdot k\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.7e-25)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (* (/ l (pow (* t_m k) 2.0)) (/ 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 <= 7.7e-25) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = (l / pow((t_m * k), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.7d-25) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else
        tmp = (l / ((t_m * k) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.7e-25) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = (l / Math.pow((t_m * k), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.7e-25:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	else:
		tmp = (l / math.pow((t_m * k), 2.0)) * (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 <= 7.7e-25)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	else
		tmp = Float64(Float64(l / (Float64(t_m * k) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.7e-25)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	else
		tmp = (l / ((t_m * k) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.7e-25], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(t$95$m * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.7000000000000002e-25

   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 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. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 63.1%

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

   10. Applied rewrites 64.1%

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

   if 7.7000000000000002e-25 < t

   1. Initial program 65.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 63.6

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

   5. Applied rewrites 63.6%

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

   7. Applied rewrites 63.6%

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 86.8%

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

Alternative 17: 65.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 (<= k 6.5e-158)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
    (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-158) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d-158) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
    else
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-158) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.5e-158:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-158)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.5e-158)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-158], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.49999999999999971e-158

   1. Initial program 54.4%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 56.6%

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

   9. Applied rewrites 59.2%

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

   11. Applied rewrites 65.1%

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

   if 6.49999999999999971e-158 < k

   1. Initial program 56.9%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.8

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

   5. Applied rewrites 60.8%

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

   7. Applied rewrites 60.8%

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

   9. Applied rewrites 60.6%

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

   11. Applied rewrites 65.8%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 61.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 58.1

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 58.1%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 59.7%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 65.7%

    \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Alternative 19: 58.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m \cdot t\_m} \end{array} \]

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 (/ (* l (/ l (* (* k k) t_m))) (* t_m t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * Float64(l / Float64(Float64(k * k) * t_m))) / Float64(t_m * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 58.1

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 58.1%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 59.7%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 60.6%

    \[\leadsto \frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{\color{blue}{t \cdot t}} \]
12. Add Preprocessing

Alternative 20: 58.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{t\_m \cdot t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (* (* k k) t_m)) (/ l (* t_m t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((k * k) * t_m)) * (l / (t_m * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / ((k * k) * t_m)) * (l / (t_m * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / ((k * k) * t_m)) * (l / (t_m * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / ((k * k) * t_m)) * (l / (t_m * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / Float64(Float64(k * k) * t_m)) * Float64(l / Float64(t_m * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / ((k * k) * t_m)) * (l / (t_m * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 58.1

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 58.1%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 59.7%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 60.5%

    \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t \cdot t}} \]
12. Add Preprocessing

Alternative 21: 53.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l l) (* (* (* k k) t_m) (* t_m t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(Float64(k * k) * t_m) * Float64(t_m * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 58.1

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 58.1%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 59.7%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 54.5%

    \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024323 
(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))))