Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 93.7%

Time: 13.0s

Alternatives: 22

Speedup: 10.7×

• 28.0% 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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(\left(\left(\cos k \cdot \ell\right) \cdot {\sin k}^{-2}\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot t\_m\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-45)
    (/ (* (* (* (* (cos k) l) (pow (sin k) -2.0)) (/ l k)) 2.0) (* t_m k))
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (* (/ t_m l) (* (* (fma (/ k t_m) (/ k t_m) 2.0) t_m) (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 <= 4.8e-45) {
		tmp = ((((cos(k) * l) * pow(sin(k), -2.0)) * (l / k)) * 2.0) / (t_m * k);
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((fma((k / t_m), (k / t_m), 2.0) * t_m) * 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 <= 4.8e-45)
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k) * l) * (sin(k) ^ -2.0)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * t_m) * 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, 4.8e-45], N[(N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.7999999999999998e-45

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 73.5%

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

   9. Applied rewrites 80.3%

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

   if 4.7999999999999998e-45 < t

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. unpow2 N/A

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

       6. times-frac N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 97.1

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

   11. Applied rewrites 97.1%

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

Alternative 2: 58.8% accurate, 0.9× speedup

Math

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

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       0.0)
    (/ (* (- l) l) (* (* (* t_m t_m) k) (* t_m k)))
    (* (/ l (* (* t_m t_m) t_m)) (/ l (* k k))))))

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 0.0d0) then
        tmp = (-l * l) / (((t_m * t_m) * k) * (t_m * k))
    else
        tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((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))) <= 0.0) {
		tmp = (-l * l) / (((t_m * t_m) * k) * (t_m * k));
	} else {
		tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (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))) <= 0.0:
		tmp = (-l * l) / (((t_m * t_m) * k) * (t_m * k))
	else:
		tmp = (l / ((t_m * t_m) * t_m)) * (l / (k * k))
	return t_s * tmp

Julia

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

MATLAB

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

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[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], 0.0], N[(N[((-l) * l), $MachinePrecision] / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $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)))) < 0.0

   1. Initial program 78.0%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 66.8

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

   5. Applied rewrites 66.8%

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 52.1%

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

   11. Applied rewrites 52.0%

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

   if 0.0 < (/.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 25.2%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 29.9

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

   5. Applied rewrites 29.9%

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

   7. Applied rewrites 29.9%

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

Alternative 3: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\left(\cos k \cdot \ell\right) \cdot \frac{{\sin k}^{-2} \cdot \ell}{t\_m \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot t\_m\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-45)
    (* (/ 2.0 k) (* (* (cos k) l) (/ (* (pow (sin k) -2.0) l) (* t_m k))))
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (* (/ t_m l) (* (* (fma (/ k t_m) (/ k t_m) 2.0) t_m) (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 <= 4.8e-45) {
		tmp = (2.0 / k) * ((cos(k) * l) * ((pow(sin(k), -2.0) * l) / (t_m * k)));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((fma((k / t_m), (k / t_m), 2.0) * t_m) * 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 <= 4.8e-45)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(cos(k) * l) * Float64(Float64((sin(k) ^ -2.0) * l) / Float64(t_m * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * t_m) * 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, 4.8e-45], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.7999999999999998e-45

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 73.5%

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

   9. Applied rewrites 79.2%

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

   if 4.7999999999999998e-45 < t

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. unpow2 N/A

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

       6. times-frac N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 97.1

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

   11. Applied rewrites 97.1%

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

Alternative 4: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{\tan k}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot t\_m\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.75e-45)
    (* (/ 2.0 k) (/ (/ (* l (/ l (sin k))) (tan k)) (* t_m k)))
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (* (/ t_m l) (* (* (fma (/ k t_m) (/ k t_m) 2.0) t_m) (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 <= 1.75e-45) {
		tmp = (2.0 / k) * (((l * (l / sin(k))) / tan(k)) / (t_m * k));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((fma((k / t_m), (k / t_m), 2.0) * t_m) * 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 <= 1.75e-45)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * Float64(l / sin(k))) / tan(k)) / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * t_m) * 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, 1.75e-45], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.75e-45

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 64.2%

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

   9. Applied rewrites 73.2%

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

   if 1.75e-45 < t

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. unpow2 N/A

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

       6. times-frac N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 97.1

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

   11. Applied rewrites 97.1%

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

Alternative 5: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{\tan k}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(2, t\_m, \frac{k \cdot k}{t\_m}\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.45e-55)
    (* (/ 2.0 k) (/ (/ (* l (/ l (sin k))) (tan k)) (* t_m k)))
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (* (/ t_m l) (* (fma 2.0 t_m (/ (* k k) t_m)) (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 <= 2.45e-55) {
		tmp = (2.0 / k) * (((l * (l / sin(k))) / tan(k)) / (t_m * k));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * (fma(2.0, t_m, ((k * k) / t_m)) * 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 <= 2.45e-55)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * Float64(l / sin(k))) / tan(k)) / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(t_m / l) * Float64(fma(2.0, t_m, Float64(Float64(k * k) / t_m)) * 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, 2.45e-55], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(2.0 * t$95$m + N[(N[(k * k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.45000000000000018e-55

   1. Initial program 44.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.3

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

   5. Applied rewrites 64.3%

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

   7. Applied rewrites 64.3%

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

   9. Applied rewrites 73.6%

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

   if 2.45000000000000018e-55 < t

   1. Initial program 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 88.3%

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

   6. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in k around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 92.2

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

   11. Applied rewrites 92.2%

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

Alternative 6: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{\left(t\_m \cdot k\right) \cdot t\_m}{\ell} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 0.145:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t\_m \cdot t\_m, 1\right)\right)}{\ell}, k \cdot k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\sin k}{\ell \cdot \ell}}}{\left(t\_m \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.5e-101)
    (/ 2.0 (* (/ (* (sin k) t_m) l) (* (/ (* (* t_m k) t_m) l) 2.0)))
    (if (<= k 0.145)
      (/
       2.0
       (*
        (* k (/ t_m l))
        (*
         (fma
          (/
           (fma
            (* (fma (* t_m t_m) 0.26666666666666666 0.3333333333333333) k)
            k
            (fma 0.6666666666666666 (* t_m t_m) 1.0))
           l)
          (* k k)
          (* (/ (* t_m t_m) l) 2.0))
         k)))
      (/ (/ 2.0 (* (tan k) (/ (sin k) (* l l)))) (* (* t_m k) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.5e-101) {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((((t_m * k) * t_m) / l) * 2.0));
	} else if (k <= 0.145) {
		tmp = 2.0 / ((k * (t_m / l)) * (fma((fma((fma((t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, (t_m * t_m), 1.0)) / l), (k * k), (((t_m * t_m) / l) * 2.0)) * k));
	} else {
		tmp = (2.0 / (tan(k) * (sin(k) / (l * l)))) / ((t_m * k) * k);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.5e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(Float64(Float64(t_m * k) * t_m) / l) * 2.0)));
	elseif (k <= 0.145)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_m / l)) * Float64(fma(Float64(fma(Float64(fma(Float64(t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, Float64(t_m * t_m), 1.0)) / l), Float64(k * k), Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) / Float64(l * l)))) / Float64(Float64(t_m * k) * k));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.5e-101], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.145], N[(2.0 / N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.26666666666666666 + 0.3333333333333333), $MachinePrecision] * k), $MachinePrecision] * k + N[(0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.4999999999999998e-101

   1. Initial program 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

   if 4.4999999999999998e-101 < k < 0.14499999999999999

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 80.0%

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 92.0%

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       3. lower-/.f64 95.9

          \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   12. Applied rewrites 95.9%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   if 0.14499999999999999 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 54.5%

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

   9. Applied rewrites 63.2%

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

Alternative 7: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\sin k}}{\tan k}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\left(2 \cdot t\_m\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-43)
    (* (/ 2.0 k) (/ (/ (* l (/ l (sin k))) (tan k)) (* t_m k)))
    (/ 2.0 (* (/ (* (sin k) t_m) l) (* (/ t_m l) (* (* 2.0 t_m) (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 <= 1.8e-43) {
		tmp = (2.0 / k) * (((l * (l / sin(k))) / tan(k)) / (t_m * k));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * tan(k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.8d-43) then
        tmp = (2.0d0 / k) * (((l * (l / sin(k))) / tan(k)) / (t_m * k))
    else
        tmp = 2.0d0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((2.0d0 * t_m) * tan(k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-43) {
		tmp = (2.0 / k) * (((l * (l / Math.sin(k))) / Math.tan(k)) / (t_m * k));
	} else {
		tmp = 2.0 / (((Math.sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * Math.tan(k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.8e-43:
		tmp = (2.0 / k) * (((l * (l / math.sin(k))) / math.tan(k)) / (t_m * k))
	else:
		tmp = 2.0 / (((math.sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * math.tan(k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-43)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * Float64(l / sin(k))) / tan(k)) / Float64(t_m * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(t_m / l) * Float64(Float64(2.0 * t_m) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.8e-43)
		tmp = (2.0 / k) * (((l * (l / sin(k))) / tan(k)) / (t_m * k));
	else
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * tan(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-43], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(2.0 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7999999999999999e-43

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 64.2%

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

   9. Applied rewrites 73.2%

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

   if 1.7999999999999999e-43 < t

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 91.4

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

   11. Applied rewrites 91.4%

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

Alternative 8: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \frac{\sin k}{\ell \cdot \ell}}}{\left(t\_m \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\left(2 \cdot t\_m\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-43)
    (/ (/ 2.0 (* (tan k) (/ (sin k) (* l l)))) (* (* t_m k) k))
    (/ 2.0 (* (/ (* (sin k) t_m) l) (* (/ t_m l) (* (* 2.0 t_m) (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 <= 1.8e-43) {
		tmp = (2.0 / (tan(k) * (sin(k) / (l * l)))) / ((t_m * k) * k);
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * tan(k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.8d-43) then
        tmp = (2.0d0 / (tan(k) * (sin(k) / (l * l)))) / ((t_m * k) * k)
    else
        tmp = 2.0d0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((2.0d0 * t_m) * tan(k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-43) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) / (l * l)))) / ((t_m * k) * k);
	} else {
		tmp = 2.0 / (((Math.sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * Math.tan(k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.8e-43:
		tmp = (2.0 / (math.tan(k) * (math.sin(k) / (l * l)))) / ((t_m * k) * k)
	else:
		tmp = 2.0 / (((math.sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * math.tan(k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-43)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) / Float64(l * l)))) / Float64(Float64(t_m * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(t_m / l) * Float64(Float64(2.0 * t_m) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.8e-43)
		tmp = (2.0 / (tan(k) * (sin(k) / (l * l)))) / ((t_m * k) * k);
	else
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((t_m / l) * ((2.0 * t_m) * tan(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-43], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(2.0 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7999999999999999e-43

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 64.2%

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

   9. Applied rewrites 69.4%

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

   if 1.7999999999999999e-43 < t

   1. Initial program 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in t around inf

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

       1. lower-*.f64 91.4

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

   11. Applied rewrites 91.4%

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

Alternative 9: 74.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{\left(t\_m \cdot k\right) \cdot t\_m}{\ell} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 0.145:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t\_m \cdot t\_m, 1\right)\right)}{\ell}, k \cdot k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.5e-101)
    (/ 2.0 (* (/ (* (sin k) t_m) l) (* (/ (* (* t_m k) t_m) l) 2.0)))
    (if (<= k 0.145)
      (/
       2.0
       (*
        (* k (/ t_m l))
        (*
         (fma
          (/
           (fma
            (* (fma (* t_m t_m) 0.26666666666666666 0.3333333333333333) k)
            k
            (fma 0.6666666666666666 (* t_m t_m) 1.0))
           l)
          (* k k)
          (* (/ (* t_m t_m) l) 2.0))
         k)))
      (/ (* (/ 2.0 (* (tan k) (sin k))) (* l l)) (* (* 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 <= 4.5e-101) {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((((t_m * k) * t_m) / l) * 2.0));
	} else if (k <= 0.145) {
		tmp = 2.0 / ((k * (t_m / l)) * (fma((fma((fma((t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, (t_m * t_m), 1.0)) / l), (k * k), (((t_m * t_m) / l) * 2.0)) * k));
	} else {
		tmp = ((2.0 / (tan(k) * sin(k))) * (l * l)) / ((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 <= 4.5e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(Float64(Float64(t_m * k) * t_m) / l) * 2.0)));
	elseif (k <= 0.145)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_m / l)) * Float64(fma(Float64(fma(Float64(fma(Float64(t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, Float64(t_m * t_m), 1.0)) / l), Float64(k * k), Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * k)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(tan(k) * sin(k))) * Float64(l * l)) / Float64(Float64(k * k) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.5e-101], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.145], N[(2.0 / N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.26666666666666666 + 0.3333333333333333), $MachinePrecision] * k), $MachinePrecision] * k + N[(0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.4999999999999998e-101

   1. Initial program 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

   if 4.4999999999999998e-101 < k < 0.14499999999999999

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 80.0%

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 92.0%

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       3. lower-/.f64 95.9

          \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   12. Applied rewrites 95.9%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   if 0.14499999999999999 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 54.5%

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

   9. Applied rewrites 54.5%

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

Alternative 10: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{\left(t\_m \cdot k\right) \cdot t\_m}{\ell} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 0.145:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t\_m \cdot t\_m, 1\right)\right)}{\ell}, k \cdot k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

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 4.5e-101)
    (/ 2.0 (* (/ (* (sin k) t_m) l) (* (/ (* (* t_m k) t_m) l) 2.0)))
    (if (<= k 0.145)
      (/
       2.0
       (*
        (* k (/ t_m l))
        (*
         (fma
          (/
           (fma
            (* (fma (* t_m t_m) 0.26666666666666666 0.3333333333333333) k)
            k
            (fma 0.6666666666666666 (* t_m t_m) 1.0))
           l)
          (* k k)
          (* (/ (* t_m t_m) l) 2.0))
         k)))
      (/ 2.0 (* (* (* k k) t_m) (* (tan k) (/ (sin k) (* l l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.5e-101) {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((((t_m * k) * t_m) / l) * 2.0));
	} else if (k <= 0.145) {
		tmp = 2.0 / ((k * (t_m / l)) * (fma((fma((fma((t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, (t_m * t_m), 1.0)) / l), (k * k), (((t_m * t_m) / l) * 2.0)) * k));
	} else {
		tmp = 2.0 / (((k * k) * t_m) * (tan(k) * (sin(k) / (l * l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.5e-101)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(Float64(Float64(t_m * k) * t_m) / l) * 2.0)));
	elseif (k <= 0.145)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_m / l)) * Float64(fma(Float64(fma(Float64(fma(Float64(t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, Float64(t_m * t_m), 1.0)) / l), Float64(k * k), Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(tan(k) * Float64(sin(k) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.5e-101], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.145], N[(2.0 / N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.26666666666666666 + 0.3333333333333333), $MachinePrecision] * k), $MachinePrecision] * k + N[(0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.4999999999999998e-101

   1. Initial program 58.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 78.6%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.7

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

   9. Applied rewrites 69.7%

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

   if 4.4999999999999998e-101 < k < 0.14499999999999999

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 80.0%

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 92.0%

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       3. lower-/.f64 95.9

          \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   12. Applied rewrites 95.9%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   if 0.14499999999999999 < k

   1. Initial program 29.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 54.4%

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

Alternative 11: 75.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t\_m \cdot t\_m, 1\right)\right)}{\ell}, k \cdot k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{\left(t\_m \cdot k\right) \cdot t\_m}{\ell} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.85e-22)
    (/
     2.0
     (*
      (* k (/ t_m l))
      (*
       (fma
        (/
         (fma
          (* (fma (* t_m t_m) 0.26666666666666666 0.3333333333333333) k)
          k
          (fma 0.6666666666666666 (* t_m t_m) 1.0))
         l)
        (* k k)
        (* (/ (* t_m t_m) l) 2.0))
       k)))
    (/ 2.0 (* (/ (* (sin k) t_m) l) (* (/ (* (* t_m k) t_m) l) 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 <= 3.85e-22) {
		tmp = 2.0 / ((k * (t_m / l)) * (fma((fma((fma((t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, (t_m * t_m), 1.0)) / l), (k * k), (((t_m * t_m) / l) * 2.0)) * k));
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((((t_m * k) * t_m) / l) * 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 <= 3.85e-22)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_m / l)) * Float64(fma(Float64(fma(Float64(fma(Float64(t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, Float64(t_m * t_m), 1.0)) / l), Float64(k * k), Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(Float64(Float64(t_m * k) * t_m) / l) * 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.85e-22], N[(2.0 / N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.26666666666666666 + 0.3333333333333333), $MachinePrecision] * k), $MachinePrecision] * k + N[(0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.8500000000000001e-22

   1. Initial program 45.3%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 62.0%

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

   6. Applied rewrites 65.7%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 61.9%

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       3. lower-/.f64 64.1

          \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   12. Applied rewrites 64.1%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   if 3.8500000000000001e-22 < t

   1. Initial program 63.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 90.2%

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

   6. Applied rewrites 91.5%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.8

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

   9. Applied rewrites 81.8%

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

Alternative 12: 75.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_m \cdot t\_m, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t\_m \cdot t\_m, 1\right)\right)}{\ell}, k \cdot k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{k}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 7e-23)
    (/
     2.0
     (*
      (* k (/ t_m l))
      (*
       (fma
        (/
         (fma
          (* (fma (* t_m t_m) 0.26666666666666666 0.3333333333333333) k)
          k
          (fma 0.6666666666666666 (* t_m t_m) 1.0))
         l)
        (* k k)
        (* (/ (* t_m t_m) l) 2.0))
       k)))
    (/ (* (/ l (* t_m k)) (/ (/ l k) 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) {
	double tmp;
	if (t_m <= 7e-23) {
		tmp = 2.0 / ((k * (t_m / l)) * (fma((fma((fma((t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, (t_m * t_m), 1.0)) / l), (k * k), (((t_m * t_m) / l) * 2.0)) * k));
	} else {
		tmp = ((l / (t_m * k)) * ((l / k) / t_m)) / 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 <= 7e-23)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_m / l)) * Float64(fma(Float64(fma(Float64(fma(Float64(t_m * t_m), 0.26666666666666666, 0.3333333333333333) * k), k, fma(0.6666666666666666, Float64(t_m * t_m), 1.0)) / l), Float64(k * k), Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * k)));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / k) / t_m)) / t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-23], N[(2.0 / N[(N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.26666666666666666 + 0.3333333333333333), $MachinePrecision] * k), $MachinePrecision] * k + N[(0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.99999999999999987e-23

   1. Initial program 45.3%

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

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. sqr-neg-rev N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 62.0%

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

   6. Applied rewrites 65.7%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 61.9%

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{4}{15}, \frac{1}{3}\right) \cdot k, k, \mathsf{fma}\left(\frac{2}{3}, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

       3. lower-/.f64 64.1

          \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   12. Applied rewrites 64.1%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.26666666666666666, 0.3333333333333333\right) \cdot k, k, \mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right)\right)}{\ell}, k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot k\right)} \]

   if 6.99999999999999987e-23 < t

   1. Initial program 63.4%

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

   3. 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 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 65.9%

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

   9. Applied rewrites 70.5%

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

   11. Applied rewrites 79.9%

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

Alternative 13: 72.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{\ell}{k}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 1.75e-44)
    (* (/ 2.0 (* (* k k) t_m)) (* (/ l k) (/ l k)))
    (/ (* (/ l (* t_m k)) (/ (/ l k) 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) {
	double tmp;
	if (t_m <= 1.75e-44) {
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	} else {
		tmp = ((l / (t_m * k)) * ((l / k) / t_m)) / 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 <= 1.75d-44) then
        tmp = (2.0d0 / ((k * k) * t_m)) * ((l / k) * (l / k))
    else
        tmp = ((l / (t_m * k)) * ((l / k) / t_m)) / 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 <= 1.75e-44) {
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	} else {
		tmp = ((l / (t_m * k)) * ((l / k) / t_m)) / 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 <= 1.75e-44:
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k))
	else:
		tmp = ((l / (t_m * k)) * ((l / k) / t_m)) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.75e-44)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(Float64(l / k) * Float64(l / k)));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(l / k) / t_m)) / 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 <= 1.75e-44)
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	else
		tmp = ((l / (t_m * k)) * ((l / k) / t_m)) / 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, 1.75e-44], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7499999999999999e-44

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 54.0%

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

   if 1.7499999999999999e-44 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 69.9%

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

   11. Applied rewrites 80.5%

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

Alternative 14: 70.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= t_m 2.4e-44)
    (* (/ 2.0 (* (* k k) t_m)) (* (/ l k) (/ l k)))
    (* (/ (/ l k) t_m) (/ l (* (* t_m t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.4e-44) {
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	} else {
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * 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 <= 2.4d-44) then
        tmp = (2.0d0 / ((k * k) * t_m)) * ((l / k) * (l / k))
    else
        tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * 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 <= 2.4e-44) {
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	} else {
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * 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 <= 2.4e-44:
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k))
	else:
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * 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 <= 2.4e-44)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(Float64(l / k) * Float64(l / k)));
	else
		tmp = Float64(Float64(Float64(l / k) / t_m) * Float64(l / Float64(Float64(t_m * t_m) * 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 <= 2.4e-44)
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	else
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * 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, 2.4e-44], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.40000000000000009e-44

   1. Initial program 45.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 54.0%

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

   if 2.40000000000000009e-44 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 69.9%

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

   11. Applied rewrites 76.4%

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

Alternative 15: 66.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \ell}{t\_m}}{\left(t\_m \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m}}{t\_m} \cdot \ell}{\left(k \cdot k\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+126)
    (/ (/ (* (/ l k) l) t_m) (* (* t_m t_m) k))
    (/ (* (/ (/ l t_m) t_m) l) (* (* 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 ((l * l) <= 5e+126) {
		tmp = (((l / k) * l) / t_m) / ((t_m * t_m) * k);
	} else {
		tmp = (((l / t_m) / t_m) * l) / ((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 ((l * l) <= 5d+126) then
        tmp = (((l / k) * l) / t_m) / ((t_m * t_m) * k)
    else
        tmp = (((l / t_m) / t_m) * l) / ((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 ((l * l) <= 5e+126) {
		tmp = (((l / k) * l) / t_m) / ((t_m * t_m) * k);
	} else {
		tmp = (((l / t_m) / t_m) * l) / ((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 (l * l) <= 5e+126:
		tmp = (((l / k) * l) / t_m) / ((t_m * t_m) * k)
	else:
		tmp = (((l / t_m) / t_m) * l) / ((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 (Float64(l * l) <= 5e+126)
		tmp = Float64(Float64(Float64(Float64(l / k) * l) / t_m) / Float64(Float64(t_m * t_m) * k));
	else
		tmp = Float64(Float64(Float64(Float64(l / t_m) / t_m) * l) / 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 ((l * l) <= 5e+126)
		tmp = (((l / k) * l) / t_m) / ((t_m * t_m) * k);
	else
		tmp = (((l / t_m) / t_m) * l) / ((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[N[(l * l), $MachinePrecision], 5e+126], N[(N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999977e126

   1. Initial program 59.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 50.8

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

   5. Applied rewrites 50.8%

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

   7. Applied rewrites 60.8%

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

   9. Applied rewrites 62.8%

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

   11. Applied rewrites 65.5%

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

   if 4.99999999999999977e126 < (*.f64 l l)

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 42.2

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

   5. Applied rewrites 42.2%

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

   7. Applied rewrites 42.2%

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

   9. Applied rewrites 53.2%

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

Alternative 16: 66.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m}}{t\_m} \cdot \ell}{\left(k \cdot k\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.2e-150)
    (* (/ (/ l k) t_m) (/ l (* (* t_m t_m) k)))
    (/ (* (/ (/ l t_m) t_m) l) (* (* 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 <= 9.2e-150) {
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k));
	} else {
		tmp = (((l / t_m) / t_m) * l) / ((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 <= 9.2d-150) then
        tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k))
    else
        tmp = (((l / t_m) / t_m) * l) / ((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 <= 9.2e-150) {
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k));
	} else {
		tmp = (((l / t_m) / t_m) * l) / ((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 <= 9.2e-150:
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k))
	else:
		tmp = (((l / t_m) / t_m) * l) / ((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 <= 9.2e-150)
		tmp = Float64(Float64(Float64(l / k) / t_m) * Float64(l / Float64(Float64(t_m * t_m) * k)));
	else
		tmp = Float64(Float64(Float64(Float64(l / t_m) / t_m) * l) / 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 <= 9.2e-150)
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k));
	else
		tmp = (((l / t_m) / t_m) * l) / ((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, 9.2e-150], N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.20000000000000011e-150

   1. Initial program 57.2%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 63.8%

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

   11. Applied rewrites 65.3%

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

   if 9.20000000000000011e-150 < k

   1. Initial program 38.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 41.2%

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

   9. Applied rewrites 47.0%

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

Alternative 17: 65.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t\_m \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t\_m} \cdot \frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= t_m 1.5e-66)
    (/ (* (/ l k) l) (* (* (* t_m k) t_m) t_m))
    (* (/ (/ l k) t_m) (/ l (* (* t_m t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-66) {
		tmp = ((l / k) * l) / (((t_m * k) * t_m) * t_m);
	} else {
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-66) then
        tmp = ((l / k) * l) / (((t_m * k) * t_m) * t_m)
    else
        tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-66) {
		tmp = ((l / k) * l) / (((t_m * k) * t_m) * t_m);
	} else {
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-66:
		tmp = ((l / k) * l) / (((t_m * k) * t_m) * t_m)
	else:
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * 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-66)
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(Float64(t_m * k) * t_m) * t_m));
	else
		tmp = Float64(Float64(Float64(l / k) / t_m) * Float64(l / Float64(Float64(t_m * t_m) * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-66)
		tmp = ((l / k) * l) / (((t_m * k) * t_m) * t_m);
	else
		tmp = ((l / k) / t_m) * (l / ((t_m * t_m) * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-66], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t$95$m * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5000000000000001e-66

   1. Initial program 44.3%

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

   3. Taylor expanded in 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 43.2

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

   5. Applied rewrites 43.2%

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

   7. Applied rewrites 47.8%

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

   9. Applied rewrites 49.5%

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

   11. Applied rewrites 50.7%

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

   if 1.5000000000000001e-66 < t

   1. Initial program 62.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 66.4%

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

   11. Applied rewrites 72.1%

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

Alternative 18: 64.3% accurate, 10.7× speedup

Math

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

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 47.2

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

5. Applied rewrites 47.2%

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

7. Applied rewrites 52.3%

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

9. Applied rewrites 54.7%

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

11. Applied rewrites 55.5%

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

Alternative 19: 63.3% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 47.2

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

5. Applied rewrites 47.2%

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

7. Applied rewrites 52.3%

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

9. Applied rewrites 54.7%

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

Alternative 20: 63.3% accurate, 10.7× speedup

Math

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

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 47.2

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

5. Applied rewrites 47.2%

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

7. Applied rewrites 52.3%

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

9. Applied rewrites 54.7%

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

Alternative 21: 34.1% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 47.2

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

5. Applied rewrites 47.2%

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

7. Applied rewrites 47.2%

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

9. Applied rewrites 26.9%

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

11. Applied rewrites 26.9%

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

Alternative 22: 34.2% accurate, 11.8× speedup

Math

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

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 (* (* t_m t_m) 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 * l) / ((k * ((t_m * t_m) * 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 * l) / ((k * ((t_m * t_m) * 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 * l) / ((k * ((t_m * t_m) * 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 * l) / ((k * ((t_m * t_m) * 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) * l) / Float64(Float64(k * Float64(Float64(t_m * t_m) * 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 * l) / ((k * ((t_m * t_m) * 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[((-l) * l), $MachinePrecision] / N[(N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
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 47.2

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

5. Applied rewrites 47.2%

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

7. Applied rewrites 47.2%

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

9. Applied rewrites 26.9%

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

11. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2024305 
(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))))