Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 95.5%

Time: 11.7s

Alternatives: 14

Speedup: 7.8×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{t\_m}{\ell}\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right) \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-61)
    (/ 2.0 (* (tan k) (* (sin k) (* (/ (* k t_m) l) (/ k l)))))
    (/
     2.0
     (*
      (* (* (tan k) (/ t_m l)) (+ (pow (/ k t_m) 2.0) 2.0))
      (* (* (sin k) t_m) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-61) {
		tmp = 2.0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))));
	} else {
		tmp = 2.0 / (((tan(k) * (t_m / l)) * (pow((k / t_m), 2.0) + 2.0)) * ((sin(k) * t_m) * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.8d-61) then
        tmp = 2.0d0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))))
    else
        tmp = 2.0d0 / (((tan(k) * (t_m / l)) * (((k / t_m) ** 2.0d0) + 2.0d0)) * ((sin(k) * t_m) * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-61) {
		tmp = 2.0 / (Math.tan(k) * (Math.sin(k) * (((k * t_m) / l) * (k / l))));
	} else {
		tmp = 2.0 / (((Math.tan(k) * (t_m / l)) * (Math.pow((k / t_m), 2.0) + 2.0)) * ((Math.sin(k) * t_m) * (t_m / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.8e-61:
		tmp = 2.0 / (math.tan(k) * (math.sin(k) * (((k * t_m) / l) * (k / l))))
	else:
		tmp = 2.0 / (((math.tan(k) * (t_m / l)) * (math.pow((k / t_m), 2.0) + 2.0)) * ((math.sin(k) * t_m) * (t_m / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-61)
		tmp = Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64(Float64(k * t_m) / l) * Float64(k / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(t_m / l)) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.8e-61)
		tmp = 2.0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))));
	else
		tmp = 2.0 / (((tan(k) * (t_m / l)) * (((k / t_m) ^ 2.0) + 2.0)) * ((sin(k) * t_m) * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-61], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-61

   1. Initial program 51.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.0

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 70.9%

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

   9. Applied rewrites 80.0%

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

   if 2.8000000000000001e-61 < t

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

   8. Applied rewrites 97.3%

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

Alternative 2: 90.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+182}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \tan k\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k\right)\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(\left(\left(\frac{\sin k}{\ell} \cdot 2\right) \cdot t\_m\right) \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-61)
    (/ 2.0 (* (tan k) (* (sin k) (* (/ (* k t_m) l) (/ k l)))))
    (if (<= t_m 3.4e+182)
      (*
       (/
        2.0
        (*
         (* t_m (tan k))
         (* (* (/ t_m l) t_m) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)))))
       l)
      (/
       2.0
       (* (/ t_m l) (* (tan k) (* (* (* (/ (sin k) l) 2.0) 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 <= 2.8e-61) {
		tmp = 2.0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))));
	} else if (t_m <= 3.4e+182) {
		tmp = (2.0 / ((t_m * tan(k)) * (((t_m / l) * t_m) * ((pow((k / t_m), 2.0) + 2.0) * sin(k))))) * l;
	} else {
		tmp = 2.0 / ((t_m / l) * (tan(k) * ((((sin(k) / l) * 2.0) * 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 <= 2.8d-61) then
        tmp = 2.0d0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))))
    else if (t_m <= 3.4d+182) then
        tmp = (2.0d0 / ((t_m * tan(k)) * (((t_m / l) * t_m) * ((((k / t_m) ** 2.0d0) + 2.0d0) * sin(k))))) * l
    else
        tmp = 2.0d0 / ((t_m / l) * (tan(k) * ((((sin(k) / l) * 2.0d0) * 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 <= 2.8e-61) {
		tmp = 2.0 / (Math.tan(k) * (Math.sin(k) * (((k * t_m) / l) * (k / l))));
	} else if (t_m <= 3.4e+182) {
		tmp = (2.0 / ((t_m * Math.tan(k)) * (((t_m / l) * t_m) * ((Math.pow((k / t_m), 2.0) + 2.0) * Math.sin(k))))) * l;
	} else {
		tmp = 2.0 / ((t_m / l) * (Math.tan(k) * ((((Math.sin(k) / l) * 2.0) * 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 <= 2.8e-61:
		tmp = 2.0 / (math.tan(k) * (math.sin(k) * (((k * t_m) / l) * (k / l))))
	elif t_m <= 3.4e+182:
		tmp = (2.0 / ((t_m * math.tan(k)) * (((t_m / l) * t_m) * ((math.pow((k / t_m), 2.0) + 2.0) * math.sin(k))))) * l
	else:
		tmp = 2.0 / ((t_m / l) * (math.tan(k) * ((((math.sin(k) / l) * 2.0) * 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 <= 2.8e-61)
		tmp = Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64(Float64(k * t_m) / l) * Float64(k / l)))));
	elseif (t_m <= 3.4e+182)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * tan(k)) * Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(tan(k) * Float64(Float64(Float64(Float64(sin(k) / l) * 2.0) * 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 <= 2.8e-61)
		tmp = 2.0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))));
	elseif (t_m <= 3.4e+182)
		tmp = (2.0 / ((t_m * tan(k)) * (((t_m / l) * t_m) * ((((k / t_m) ^ 2.0) + 2.0) * sin(k))))) * l;
	else
		tmp = 2.0 / ((t_m / l) * (tan(k) * ((((sin(k) / l) * 2.0) * 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, 2.8e-61], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+182], N[(N[(2.0 / N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8000000000000001e-61

   1. Initial program 51.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.0

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 70.9%

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

   9. Applied rewrites 80.0%

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

   if 2.8000000000000001e-61 < t < 3.39999999999999987e182

   1. Initial program 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 92.4%

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

   if 3.39999999999999987e182 < t

   1. Initial program 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 94.7%

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

   7. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 94.8

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

   9. Applied rewrites 94.8%

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

Alternative 3: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 2.8e-61)
    (/ 2.0 (* (tan k) (* (sin k) (* (/ (* k t_m) l) (/ k l)))))
    (/
     2.0
     (*
      (*
       (* (tan k) (/ t_m l))
       (* (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k)) (/ t_m l)))
      t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-61) {
		tmp = 2.0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))));
	} else {
		tmp = 2.0 / (((tan(k) * (t_m / l)) * (((pow((k / t_m), 2.0) + 2.0) * sin(k)) * (t_m / l))) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.8d-61) then
        tmp = 2.0d0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))))
    else
        tmp = 2.0d0 / (((tan(k) * (t_m / l)) * (((((k / t_m) ** 2.0d0) + 2.0d0) * sin(k)) * (t_m / l))) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-61) {
		tmp = 2.0 / (Math.tan(k) * (Math.sin(k) * (((k * t_m) / l) * (k / l))));
	} else {
		tmp = 2.0 / (((Math.tan(k) * (t_m / l)) * (((Math.pow((k / t_m), 2.0) + 2.0) * Math.sin(k)) * (t_m / l))) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.8e-61:
		tmp = 2.0 / (math.tan(k) * (math.sin(k) * (((k * t_m) / l) * (k / l))))
	else:
		tmp = 2.0 / (((math.tan(k) * (t_m / l)) * (((math.pow((k / t_m), 2.0) + 2.0) * math.sin(k)) * (t_m / l))) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-61)
		tmp = Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(Float64(Float64(k * t_m) / l) * Float64(k / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(t_m / l)) * Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k)) * Float64(t_m / l))) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.8e-61)
		tmp = 2.0 / (tan(k) * (sin(k) * (((k * t_m) / l) * (k / l))));
	else
		tmp = 2.0 / (((tan(k) * (t_m / l)) * (((((k / t_m) ^ 2.0) + 2.0) * sin(k)) * (t_m / l))) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-61], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-61

   1. Initial program 51.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.0

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 70.9%

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

   9. Applied rewrites 80.0%

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

   if 2.8000000000000001e-61 < t

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 93.8%

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

Alternative 4: 87.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(\left(\left(\frac{\sin k}{\ell} \cdot 2\right) \cdot t\_m\right) \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e+51)
    (/ 2.0 (* (sin k) (* (tan k) (* (/ (* k t_m) l) (/ k l)))))
    (/ 2.0 (* (/ t_m l) (* (tan k) (* (* (* (/ (sin k) l) 2.0) 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 <= 1.25e+51) {
		tmp = 2.0 / (sin(k) * (tan(k) * (((k * t_m) / l) * (k / l))));
	} else {
		tmp = 2.0 / ((t_m / l) * (tan(k) * ((((sin(k) / l) * 2.0) * 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 <= 1.25d+51) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * (((k * t_m) / l) * (k / l))))
    else
        tmp = 2.0d0 / ((t_m / l) * (tan(k) * ((((sin(k) / l) * 2.0d0) * 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 <= 1.25e+51) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (((k * t_m) / l) * (k / l))));
	} else {
		tmp = 2.0 / ((t_m / l) * (Math.tan(k) * ((((Math.sin(k) / l) * 2.0) * 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 <= 1.25e+51:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (((k * t_m) / l) * (k / l))))
	else:
		tmp = 2.0 / ((t_m / l) * (math.tan(k) * ((((math.sin(k) / l) * 2.0) * 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 <= 1.25e+51)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(Float64(k * t_m) / l) * Float64(k / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(tan(k) * Float64(Float64(Float64(Float64(sin(k) / l) * 2.0) * 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 <= 1.25e+51)
		tmp = 2.0 / (sin(k) * (tan(k) * (((k * t_m) / l) * (k / l))));
	else
		tmp = 2.0 / ((t_m / l) * (tan(k) * ((((sin(k) / l) * 2.0) * 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, 1.25e+51], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e51

   1. Initial program 55.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 72.5%

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

   9. Applied rewrites 80.3%

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

   if 1.25e51 < t

   1. Initial program 60.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 88.2

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

   9. Applied rewrites 88.2%

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

Alternative 5: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{\left(k \cdot 2\right) \cdot t\_m}{\frac{\ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 1.2e-45)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* (* k 2.0) t_m) (/ l t_m))))
    (/ 2.0 (* (sin k) (* (tan k) (* (/ (* k t_m) l) (/ k l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-45) {
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (((k * t_m) / l) * (k / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d-45) then
        tmp = 2.0d0 / ((k / (l / t_m)) * (((k * 2.0d0) * t_m) / (l / t_m)))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * (((k * t_m) / l) * (k / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-45) {
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (((k * t_m) / l) * (k / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.2e-45:
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (((k * t_m) / l) * (k / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.2e-45)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(Float64(k * 2.0) * t_m) / Float64(l / t_m))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(Float64(k * t_m) / l) * Float64(k / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.2e-45)
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * (((k * t_m) / l) * (k / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.2e-45], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999995e-45

   1. Initial program 61.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 60.7

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

   5. Applied rewrites 60.7%

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

   7. Applied rewrites 56.5%

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

   9. Applied rewrites 79.6%

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

   if 1.19999999999999995e-45 < k

   1. Initial program 44.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 69.3

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

   5. Applied rewrites 69.3%

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

   7. Applied rewrites 68.0%

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

   9. Applied rewrites 82.1%

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

Alternative 6: 77.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(\mathsf{fma}\left(1 + -0.3333333333333333 \cdot \left(t\_m \cdot t\_m\right), \frac{k \cdot k}{\ell}, \left(t\_m \cdot \frac{t\_m}{\ell}\right) \cdot 2\right) \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{\left(k \cdot 2\right) \cdot t\_m}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.3e-36)
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (tan k)
       (*
        (fma
         (+ 1.0 (* -0.3333333333333333 (* t_m t_m)))
         (/ (* k k) l)
         (* (* t_m (/ t_m l)) 2.0))
        k))))
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* (* k 2.0) t_m) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.3e-36) {
		tmp = 2.0 / ((t_m / l) * (tan(k) * (fma((1.0 + (-0.3333333333333333 * (t_m * t_m))), ((k * k) / l), ((t_m * (t_m / l)) * 2.0)) * k)));
	} else {
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.3e-36)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(tan(k) * Float64(fma(Float64(1.0 + Float64(-0.3333333333333333 * Float64(t_m * t_m))), Float64(Float64(k * k) / l), Float64(Float64(t_m * Float64(t_m / l)) * 2.0)) * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(Float64(k * 2.0) * t_m) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.3e-36], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(1.0 + N[(-0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2999999999999998e-36

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 59.7%

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

   if 5.2999999999999998e-36 < t

   1. Initial program 65.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 54.0%

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

   9. Applied rewrites 78.5%

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

Alternative 7: 76.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{\left(k \cdot 2\right) \cdot t\_m}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-45)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* (* k 2.0) t_m) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.35d-45) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else
        tmp = 2.0d0 / ((k / (l / t_m)) * (((k * 2.0d0) * t_m) / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.35e-45:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	else:
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(Float64(k * 2.0) * t_m) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.35e-45)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	else
		tmp = 2.0 / ((k / (l / t_m)) * (((k * 2.0) * t_m) / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-45], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999992e-45

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 60.9%

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

   if 1.34999999999999992e-45 < t

   1. Initial program 66.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 55.3%

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

   9. Applied rewrites 79.0%

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

Alternative 8: 66.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_m \cdot k\right) \cdot \left(2 \cdot k\right)\right) \cdot \left(t\_m \cdot t\_m\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 2.3e-30)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (if (<= t_m 3.4e+149)
      (/ 2.0 (/ (* (* (* t_m k) (* 2.0 k)) (* t_m t_m)) (* l l)))
      (/ 2.0 (* (* (* k k) 2.0) (* t_m (* (/ t_m l) (/ t_m l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-30) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else if (t_m <= 3.4e+149) {
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * (t_m * ((t_m / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-30) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else if (t_m <= 3.4d+149) then
        tmp = 2.0d0 / ((((t_m * k) * (2.0d0 * k)) * (t_m * t_m)) / (l * l))
    else
        tmp = 2.0d0 / (((k * k) * 2.0d0) * (t_m * ((t_m / l) * (t_m / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-30) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else if (t_m <= 3.4e+149) {
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * (t_m * ((t_m / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-30:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	elif t_m <= 3.4e+149:
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l))
	else:
		tmp = 2.0 / (((k * k) * 2.0) * (t_m * ((t_m / l) * (t_m / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-30)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	elseif (t_m <= 3.4e+149)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * k) * Float64(2.0 * k)) * Float64(t_m * t_m)) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * 2.0) * Float64(t_m * Float64(Float64(t_m / l) * Float64(t_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-30)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	elseif (t_m <= 3.4e+149)
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	else
		tmp = 2.0 / (((k * k) * 2.0) * (t_m * ((t_m / l) * (t_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-30], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+149], N[(2.0 / N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.29999999999999984e-30

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.4%

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

   10. Applied rewrites 61.4%

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

   if 2.29999999999999984e-30 < t < 3.3999999999999998e149

   1. Initial program 68.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 52.8

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

   5. Applied rewrites 52.8%

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

   7. Applied rewrites 54.7%

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

   9. Applied rewrites 58.9%

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

   11. Applied rewrites 63.8%

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

   if 3.3999999999999998e149 < t

   1. Initial program 62.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 62.1

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

   5. Applied rewrites 62.1%

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

   7. Applied rewrites 51.7%

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

   9. Applied rewrites 68.9%

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

Alternative 9: 72.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\left(\left(t\_m \cdot \frac{t\_m}{\ell}\right) \cdot k\right) \cdot 2\right) \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-45)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (/ 2.0 (* (/ t_m l) (* (* (* (* t_m (/ t_m l)) k) 2.0) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / ((t_m / l) * ((((t_m * (t_m / l)) * k) * 2.0) * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.35d-45) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else
        tmp = 2.0d0 / ((t_m / l) * ((((t_m * (t_m / l)) * k) * 2.0d0) * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / ((t_m / l) * ((((t_m * (t_m / l)) * k) * 2.0) * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.35e-45:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	else:
		tmp = 2.0 / ((t_m / l) * ((((t_m * (t_m / l)) * k) * 2.0) * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(Float64(Float64(t_m * Float64(t_m / l)) * k) * 2.0) * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.35e-45)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	else
		tmp = 2.0 / ((t_m / l) * ((((t_m * (t_m / l)) * k) * 2.0) * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-45], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999992e-45

   1. Initial program 51.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.4

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 60.9%

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

   if 1.34999999999999992e-45 < t

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 93.5%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 75.3

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

   9. Applied rewrites 75.3%

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

Alternative 10: 66.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.2e-36)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (/ 2.0 (* (/ (* (* (* k k) 2.0) t_m) l) (* (/ t_m l) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.2e-36) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) / l) * ((t_m / l) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.2d-36) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else
        tmp = 2.0d0 / (((((k * k) * 2.0d0) * t_m) / l) * ((t_m / l) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.2e-36) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) / l) * ((t_m / l) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.2e-36:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	else:
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) / l) * ((t_m / l) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.2e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) / l) * Float64(Float64(t_m / l) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.2e-36)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	else
		tmp = 2.0 / (((((k * k) * 2.0) * t_m) / l) * ((t_m / l) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.2e-36], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.20000000000000025e-36

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.5%

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

   10. Applied rewrites 61.5%

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

   if 8.20000000000000025e-36 < t

   1. Initial program 65.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 63.4%

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

Alternative 11: 65.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_m \cdot k\right) \cdot \left(2 \cdot k\right)\right) \cdot \left(t\_m \cdot t\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-30)
    (/ 2.0 (* (* (* k k) (/ (* k k) l)) (/ t_m l)))
    (/ 2.0 (/ (* (* (* t_m k) (* 2.0 k)) (* t_m t_m)) (* l l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-30) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-30) then
        tmp = 2.0d0 / (((k * k) * ((k * k) / l)) * (t_m / l))
    else
        tmp = 2.0d0 / ((((t_m * k) * (2.0d0 * k)) * (t_m * t_m)) / (l * l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-30) {
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	} else {
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-30:
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l))
	else:
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-30)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * Float64(Float64(k * k) / l)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * k) * Float64(2.0 * k)) * Float64(t_m * t_m)) / Float64(l * l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-30)
		tmp = 2.0 / (((k * k) * ((k * k) / l)) * (t_m / l));
	else
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-30], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999984e-30

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.4%

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

   10. Applied rewrites 61.4%

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

   if 2.29999999999999984e-30 < t

   1. Initial program 65.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.6

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 53.5%

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

   9. Applied rewrites 55.9%

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

   11. Applied rewrites 61.7%

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

Alternative 12: 61.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_m \cdot k\right) \cdot \left(2 \cdot k\right)\right) \cdot \left(t\_m \cdot t\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.2e-36)
    (/ 2.0 (/ (* (* (* k k) t_m) (* k k)) (* l l)))
    (/ 2.0 (/ (* (* (* t_m k) (* 2.0 k)) (* t_m t_m)) (* l l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.2e-36) {
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l));
	} else {
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.2d-36) then
        tmp = 2.0d0 / ((((k * k) * t_m) * (k * k)) / (l * l))
    else
        tmp = 2.0d0 / ((((t_m * k) * (2.0d0 * k)) * (t_m * t_m)) / (l * l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.2e-36) {
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l));
	} else {
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.2e-36:
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l))
	else:
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.2e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) * Float64(k * k)) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * k) * Float64(2.0 * k)) * Float64(t_m * t_m)) / Float64(l * l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.2e-36)
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l));
	else
		tmp = 2.0 / ((((t_m * k) * (2.0 * k)) * (t_m * t_m)) / (l * l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.2e-36], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * k), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.20000000000000025e-36

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.5%

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

   10. Applied rewrites 53.3%

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

   12. Applied rewrites 53.3%

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

   if 8.20000000000000025e-36 < t

   1. Initial program 65.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 53.4%

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

   9. Applied rewrites 55.8%

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

   11. Applied rewrites 61.3%

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

Alternative 13: 57.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.95 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot t\_m\right) \cdot \frac{\left(\left(k \cdot k\right) \cdot 2\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.95e-34)
    (/ 2.0 (/ (* (* (* k k) t_m) (* k k)) (* l l)))
    (/ 2.0 (* (* t_m t_m) (/ (* (* (* k k) 2.0) t_m) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.95e-34) {
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l));
	} else {
		tmp = 2.0 / ((t_m * t_m) * ((((k * k) * 2.0) * t_m) / (l * l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.95d-34) then
        tmp = 2.0d0 / ((((k * k) * t_m) * (k * k)) / (l * l))
    else
        tmp = 2.0d0 / ((t_m * t_m) * ((((k * k) * 2.0d0) * t_m) / (l * l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.95e-34) {
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l));
	} else {
		tmp = 2.0 / ((t_m * t_m) * ((((k * k) * 2.0) * t_m) / (l * l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.95e-34:
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l))
	else:
		tmp = 2.0 / ((t_m * t_m) * ((((k * k) * 2.0) * t_m) / (l * l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.95e-34)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_m) * Float64(k * k)) / Float64(l * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * t_m) * Float64(Float64(Float64(Float64(k * k) * 2.0) * t_m) / Float64(l * l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.95e-34)
		tmp = 2.0 / ((((k * k) * t_m) * (k * k)) / (l * l));
	else
		tmp = 2.0 / ((t_m * t_m) * ((((k * k) * 2.0) * t_m) / (l * l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.95e-34], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.9500000000000001e-34

   1. Initial program 52.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-cos.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.5%

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

   10. Applied rewrites 53.3%

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

   12. Applied rewrites 53.3%

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

   if 2.9500000000000001e-34 < t

   1. Initial program 65.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 57.9%

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

   9. Applied rewrites 56.1%

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

Alternative 14: 52.3% accurate, 9.6× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in t around 0

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

   1. associate-/l* N/A

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

   2. times-frac N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-cos.f64 68.0

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

5. Applied rewrites 68.0%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 56.4%

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

10. Applied rewrites 50.2%

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

12. Applied rewrites 50.6%

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

Reproduce

herbie shell --seed 2024315 
(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))))