Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 90.5%

Time: 12.1s

Alternatives: 16

Speedup: 10.7×

• 27.8% 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: 54.4% 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: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.9:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\left(t\_m \cdot \frac{k}{\ell}\right) \cdot k\right)}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\frac{\left({t\_m}^{1.5} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot {t\_m}^{1.5}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\_m\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t\_m}\right)}^{2}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.9)
    (/ 2.0 (/ (* (pow (sin k) 2.0) (* (* t_m (/ k l)) k)) (* (cos k) l)))
    (if (<= t_m 1.2e+193)
      (/
       2.0
       (/
        (*
         (* (pow t_m 1.5) (/ (sin k) l))
         (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (pow t_m 1.5)))
        l))
      (* (pow (* k t_m) -1.0) (pow (/ k (pow (/ l t_m) 2.0)) -1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.9) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	} else if (t_m <= 1.2e+193) {
		tmp = 2.0 / (((pow(t_m, 1.5) * (sin(k) / l)) * (((pow((k / t_m), 2.0) + 2.0) * tan(k)) * pow(t_m, 1.5))) / l);
	} else {
		tmp = pow((k * t_m), -1.0) * pow((k / pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 0.9d0) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((t_m * (k / l)) * k)) / (cos(k) * l))
    else if (t_m <= 1.2d+193) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) * (sin(k) / l)) * (((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * (t_m ** 1.5d0))) / l)
    else
        tmp = ((k * t_m) ** (-1.0d0)) * ((k / ((l / t_m) ** 2.0d0)) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.9) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (Math.cos(k) * l));
	} else if (t_m <= 1.2e+193) {
		tmp = 2.0 / (((Math.pow(t_m, 1.5) * (Math.sin(k) / l)) * (((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * Math.pow(t_m, 1.5))) / l);
	} else {
		tmp = Math.pow((k * t_m), -1.0) * Math.pow((k / Math.pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 0.9:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (math.cos(k) * l))
	elif t_m <= 1.2e+193:
		tmp = 2.0 / (((math.pow(t_m, 1.5) * (math.sin(k) / l)) * (((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * math.pow(t_m, 1.5))) / l)
	else:
		tmp = math.pow((k * t_m), -1.0) * math.pow((k / math.pow((l / t_m), 2.0)), -1.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.9)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(t_m * Float64(k / l)) * k)) / Float64(cos(k) * l)));
	elseif (t_m <= 1.2e+193)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 1.5) * Float64(sin(k) / l)) * Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * (t_m ^ 1.5))) / l));
	else
		tmp = Float64((Float64(k * t_m) ^ -1.0) * (Float64(k / (Float64(l / t_m) ^ 2.0)) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 0.9)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	elseif (t_m <= 1.2e+193)
		tmp = 2.0 / ((((t_m ^ 1.5) * (sin(k) / l)) * (((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * (t_m ^ 1.5))) / l);
	else
		tmp = ((k * t_m) ^ -1.0) * ((k / ((l / t_m) ^ 2.0)) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.9], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+193], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(k * t$95$m), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(k / N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 0.900000000000000022

   1. Initial program 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 15.9

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 15.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 79.6%

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

   11. Applied rewrites 82.8%

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

   if 0.900000000000000022 < t < 1.2e193

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 88.6

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 88.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 93.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot {t}^{1.5}\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 95.2%

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

   if 1.2e193 < t

   1. Initial program 72.4%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 64.2%

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

   9. Applied rewrites 64.2%

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

   11. Applied rewrites 92.2%

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

Alternative 2: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.9:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\left(t\_m \cdot \frac{k}{\ell}\right) \cdot k\right)}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot {t\_m}^{1.5}\right) \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\_m\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t\_m}\right)}^{2}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.9)
    (/ 2.0 (/ (* (pow (sin k) 2.0) (* (* t_m (/ k l)) k)) (* (cos k) l)))
    (if (<= t_m 9e+193)
      (/
       2.0
       (*
        (* (/ (sin k) l) (pow t_m 1.5))
        (* (/ (pow t_m 1.5) l) (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0)))))
      (* (pow (* k t_m) -1.0) (pow (/ k (pow (/ l t_m) 2.0)) -1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.9) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	} else if (t_m <= 9e+193) {
		tmp = 2.0 / (((sin(k) / l) * pow(t_m, 1.5)) * ((pow(t_m, 1.5) / l) * (tan(k) * (pow((k / t_m), 2.0) + 2.0))));
	} else {
		tmp = pow((k * t_m), -1.0) * pow((k / pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 0.9d0) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((t_m * (k / l)) * k)) / (cos(k) * l))
    else if (t_m <= 9d+193) then
        tmp = 2.0d0 / (((sin(k) / l) * (t_m ** 1.5d0)) * (((t_m ** 1.5d0) / l) * (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))))
    else
        tmp = ((k * t_m) ** (-1.0d0)) * ((k / ((l / t_m) ** 2.0d0)) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.9) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (Math.cos(k) * l));
	} else if (t_m <= 9e+193) {
		tmp = 2.0 / (((Math.sin(k) / l) * Math.pow(t_m, 1.5)) * ((Math.pow(t_m, 1.5) / l) * (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0))));
	} else {
		tmp = Math.pow((k * t_m), -1.0) * Math.pow((k / Math.pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 0.9:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (math.cos(k) * l))
	elif t_m <= 9e+193:
		tmp = 2.0 / (((math.sin(k) / l) * math.pow(t_m, 1.5)) * ((math.pow(t_m, 1.5) / l) * (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))))
	else:
		tmp = math.pow((k * t_m), -1.0) * math.pow((k / math.pow((l / t_m), 2.0)), -1.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.9)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(t_m * Float64(k / l)) * k)) / Float64(cos(k) * l)));
	elseif (t_m <= 9e+193)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * (t_m ^ 1.5)) * Float64(Float64((t_m ^ 1.5) / l) * Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)))));
	else
		tmp = Float64((Float64(k * t_m) ^ -1.0) * (Float64(k / (Float64(l / t_m) ^ 2.0)) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 0.9)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	elseif (t_m <= 9e+193)
		tmp = 2.0 / (((sin(k) / l) * (t_m ^ 1.5)) * (((t_m ^ 1.5) / l) * (tan(k) * (((k / t_m) ^ 2.0) + 2.0))));
	else
		tmp = ((k * t_m) ^ -1.0) * ((k / ((l / t_m) ^ 2.0)) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.9], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+193], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(k * t$95$m), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(k / N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 0.900000000000000022

   1. Initial program 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 15.9

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 15.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 79.6%

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

   11. Applied rewrites 82.8%

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

   if 0.900000000000000022 < t < 8.99999999999999999e193

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 88.6

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 88.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 93.8%

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

   if 8.99999999999999999e193 < t

   1. Initial program 72.4%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 64.2%

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

   9. Applied rewrites 64.2%

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

   11. Applied rewrites 92.2%

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

Alternative 3: 89.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.9:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\left(t\_m \cdot \frac{k}{\ell}\right) \cdot k\right)}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \frac{\left(t\_m \cdot t\_m\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\_m\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t\_m}\right)}^{2}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.9)
    (/ 2.0 (/ (* (pow (sin k) 2.0) (* (* t_m (/ k l)) k)) (* (cos k) l)))
    (if (<= t_m 5.8e+125)
      (/
       2.0
       (*
        (* (* (/ t_m l) (/ (* (* t_m t_m) (sin k)) l)) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (* (pow (* k t_m) -1.0) (pow (/ k (pow (/ l t_m) 2.0)) -1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.9) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	} else if (t_m <= 5.8e+125) {
		tmp = 2.0 / ((((t_m / l) * (((t_m * t_m) * sin(k)) / l)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = pow((k * t_m), -1.0) * pow((k / pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 0.9d0) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((t_m * (k / l)) * k)) / (cos(k) * l))
    else if (t_m <= 5.8d+125) then
        tmp = 2.0d0 / ((((t_m / l) * (((t_m * t_m) * sin(k)) / l)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = ((k * t_m) ** (-1.0d0)) * ((k / ((l / t_m) ** 2.0d0)) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.9) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (Math.cos(k) * l));
	} else if (t_m <= 5.8e+125) {
		tmp = 2.0 / ((((t_m / l) * (((t_m * t_m) * Math.sin(k)) / l)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = Math.pow((k * t_m), -1.0) * Math.pow((k / Math.pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 0.9:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (math.cos(k) * l))
	elif t_m <= 5.8e+125:
		tmp = 2.0 / ((((t_m / l) * (((t_m * t_m) * math.sin(k)) / l)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = math.pow((k * t_m), -1.0) * math.pow((k / math.pow((l / t_m), 2.0)), -1.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.9)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(t_m * Float64(k / l)) * k)) / Float64(cos(k) * l)));
	elseif (t_m <= 5.8e+125)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(Float64(Float64(t_m * t_m) * sin(k)) / l)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64((Float64(k * t_m) ^ -1.0) * (Float64(k / (Float64(l / t_m) ^ 2.0)) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 0.9)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	elseif (t_m <= 5.8e+125)
		tmp = 2.0 / ((((t_m / l) * (((t_m * t_m) * sin(k)) / l)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = ((k * t_m) ^ -1.0) * ((k / ((l / t_m) ^ 2.0)) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.9], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+125], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(k * t$95$m), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(k / N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 0.900000000000000022

   1. Initial program 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 15.9

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 15.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 79.6%

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

   11. Applied rewrites 82.8%

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

   if 0.900000000000000022 < t < 5.79999999999999986e125

   1. Initial program 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

   if 5.79999999999999986e125 < t

   1. Initial program 71.2%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 63.8%

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

   9. Applied rewrites 66.2%

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

   11. Applied rewrites 88.3%

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

Alternative 4: 87.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\ell}{k}}{{t\_m}^{1.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\left(t\_m \cdot \frac{k}{\ell}\right) \cdot k\right)}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+171}:\\ \;\;\;\;t\_2 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\_m\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t\_m}\right)}^{2}}\right)}^{-1}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ l k) (pow t_m 1.5))))
   (*
    t_s
    (if (<= t_m 1.0)
      (/ 2.0 (/ (* (pow (sin k) 2.0) (* (* t_m (/ k l)) k)) (* (cos k) l)))
      (if (<= t_m 9e+171)
        (* t_2 t_2)
        (* (pow (* k t_m) -1.0) (pow (/ k (pow (/ l t_m) 2.0)) -1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / k) / pow(t_m, 1.5);
	double tmp;
	if (t_m <= 1.0) {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	} else if (t_m <= 9e+171) {
		tmp = t_2 * t_2;
	} else {
		tmp = pow((k * t_m), -1.0) * pow((k / pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / k) / (t_m ** 1.5d0)
    if (t_m <= 1.0d0) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((t_m * (k / l)) * k)) / (cos(k) * l))
    else if (t_m <= 9d+171) then
        tmp = t_2 * t_2
    else
        tmp = ((k * t_m) ** (-1.0d0)) * ((k / ((l / t_m) ** 2.0d0)) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / k) / Math.pow(t_m, 1.5);
	double tmp;
	if (t_m <= 1.0) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (Math.cos(k) * l));
	} else if (t_m <= 9e+171) {
		tmp = t_2 * t_2;
	} else {
		tmp = Math.pow((k * t_m), -1.0) * Math.pow((k / Math.pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (l / k) / math.pow(t_m, 1.5)
	tmp = 0
	if t_m <= 1.0:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((t_m * (k / l)) * k)) / (math.cos(k) * l))
	elif t_m <= 9e+171:
		tmp = t_2 * t_2
	else:
		tmp = math.pow((k * t_m), -1.0) * math.pow((k / math.pow((l / t_m), 2.0)), -1.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / k) / (t_m ^ 1.5))
	tmp = 0.0
	if (t_m <= 1.0)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(t_m * Float64(k / l)) * k)) / Float64(cos(k) * l)));
	elseif (t_m <= 9e+171)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64((Float64(k * t_m) ^ -1.0) * (Float64(k / (Float64(l / t_m) ^ 2.0)) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l / k) / (t_m ^ 1.5);
	tmp = 0.0;
	if (t_m <= 1.0)
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((t_m * (k / l)) * k)) / (cos(k) * l));
	elseif (t_m <= 9e+171)
		tmp = t_2 * t_2;
	else
		tmp = ((k * t_m) ^ -1.0) * ((k / ((l / t_m) ^ 2.0)) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.0], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+171], N[(t$95$2 * t$95$2), $MachinePrecision], N[(N[Power[N[(k * t$95$m), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(k / N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1

   1. Initial program 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 15.9

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 15.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 79.6%

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

   11. Applied rewrites 82.8%

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

   if 1 < t < 8.99999999999999937e171

   1. Initial program 74.0%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 73.5%

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

   9. Applied rewrites 76.4%

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

   11. Applied rewrites 93.3%

       \[\leadsto \frac{\frac{-\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{-\ell}{k}}{{t}^{1.5}}} \]

   if 8.99999999999999937e171 < t

   1. Initial program 71.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 61.8%

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

   9. Applied rewrites 61.8%

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

   11. Applied rewrites 87.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot k\right)}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \frac{\frac{\ell}{k}}{{t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t}\right)}^{2}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\ell}{k}}{{t\_m}^{1.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2300000:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t\_m}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+171}:\\ \;\;\;\;t\_2 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\_m\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t\_m}\right)}^{2}}\right)}^{-1}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ l k) (pow t_m 1.5))))
   (*
    t_s
    (if (<= t_m 2300000.0)
      (/ 2.0 (* (/ (* (* (/ k l) k) t_m) l) (* (tan k) (sin k))))
      (if (<= t_m 9e+171)
        (* t_2 t_2)
        (* (pow (* k t_m) -1.0) (pow (/ k (pow (/ l t_m) 2.0)) -1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / k) / pow(t_m, 1.5);
	double tmp;
	if (t_m <= 2300000.0) {
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)));
	} else if (t_m <= 9e+171) {
		tmp = t_2 * t_2;
	} else {
		tmp = pow((k * t_m), -1.0) * pow((k / pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / k) / (t_m ** 1.5d0)
    if (t_m <= 2300000.0d0) then
        tmp = 2.0d0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)))
    else if (t_m <= 9d+171) then
        tmp = t_2 * t_2
    else
        tmp = ((k * t_m) ** (-1.0d0)) * ((k / ((l / t_m) ** 2.0d0)) ** (-1.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / k) / Math.pow(t_m, 1.5);
	double tmp;
	if (t_m <= 2300000.0) {
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (Math.tan(k) * Math.sin(k)));
	} else if (t_m <= 9e+171) {
		tmp = t_2 * t_2;
	} else {
		tmp = Math.pow((k * t_m), -1.0) * Math.pow((k / Math.pow((l / t_m), 2.0)), -1.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (l / k) / math.pow(t_m, 1.5)
	tmp = 0
	if t_m <= 2300000.0:
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (math.tan(k) * math.sin(k)))
	elif t_m <= 9e+171:
		tmp = t_2 * t_2
	else:
		tmp = math.pow((k * t_m), -1.0) * math.pow((k / math.pow((l / t_m), 2.0)), -1.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / k) / (t_m ^ 1.5))
	tmp = 0.0
	if (t_m <= 2300000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * k) * t_m) / l) * Float64(tan(k) * sin(k))));
	elseif (t_m <= 9e+171)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64((Float64(k * t_m) ^ -1.0) * (Float64(k / (Float64(l / t_m) ^ 2.0)) ^ -1.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l / k) / (t_m ^ 1.5);
	tmp = 0.0;
	if (t_m <= 2300000.0)
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)));
	elseif (t_m <= 9e+171)
		tmp = t_2 * t_2;
	else
		tmp = ((k * t_m) ^ -1.0) * ((k / ((l / t_m) ^ 2.0)) ^ -1.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2300000.0], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+171], N[(t$95$2 * t$95$2), $MachinePrecision], N[(N[Power[N[(k * t$95$m), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(k / N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3e6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 17.7

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 17.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 78.6%

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

   11. Applied rewrites 81.5%

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

   if 2.3e6 < t < 8.99999999999999937e171

   1. Initial program 71.4%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 76.4%

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

   11. Applied rewrites 92.7%

       \[\leadsto \frac{\frac{-\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{-\ell}{k}}{{t}^{1.5}}} \]

   if 8.99999999999999937e171 < t

   1. Initial program 71.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 61.8%

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

   9. Applied rewrites 61.8%

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

   11. Applied rewrites 87.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2300000:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \frac{\frac{\ell}{k}}{{t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot t\right)}^{-1} \cdot {\left(\frac{k}{{\left(\frac{\ell}{t}\right)}^{2}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\ell}{k}}{{t\_m}^{1.5}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2300000:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t\_m}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t\_m \leq 1.75 \cdot 10^{+187}:\\ \;\;\;\;t\_2 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ l k) (pow t_m 1.5))))
   (*
    t_s
    (if (<= t_m 2300000.0)
      (/ 2.0 (* (/ (* (* (/ k l) k) t_m) l) (* (tan k) (sin k))))
      (if (<= t_m 1.75e+187)
        (* t_2 t_2)
        (/ (pow (/ l t_m) 2.0) (* (* k t_m) k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / k) / pow(t_m, 1.5);
	double tmp;
	if (t_m <= 2300000.0) {
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)));
	} else if (t_m <= 1.75e+187) {
		tmp = t_2 * t_2;
	} else {
		tmp = pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / k) / (t_m ** 1.5d0)
    if (t_m <= 2300000.0d0) then
        tmp = 2.0d0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)))
    else if (t_m <= 1.75d+187) then
        tmp = t_2 * t_2
    else
        tmp = ((l / t_m) ** 2.0d0) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (l / k) / Math.pow(t_m, 1.5);
	double tmp;
	if (t_m <= 2300000.0) {
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (Math.tan(k) * Math.sin(k)));
	} else if (t_m <= 1.75e+187) {
		tmp = t_2 * t_2;
	} else {
		tmp = Math.pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (l / k) / math.pow(t_m, 1.5)
	tmp = 0
	if t_m <= 2300000.0:
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (math.tan(k) * math.sin(k)))
	elif t_m <= 1.75e+187:
		tmp = t_2 * t_2
	else:
		tmp = math.pow((l / t_m), 2.0) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / k) / (t_m ^ 1.5))
	tmp = 0.0
	if (t_m <= 2300000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * k) * t_m) / l) * Float64(tan(k) * sin(k))));
	elseif (t_m <= 1.75e+187)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64((Float64(l / t_m) ^ 2.0) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (l / k) / (t_m ^ 1.5);
	tmp = 0.0;
	if (t_m <= 2300000.0)
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)));
	elseif (t_m <= 1.75e+187)
		tmp = t_2 * t_2;
	else
		tmp = ((l / t_m) ^ 2.0) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2300000.0], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.75e+187], N[(t$95$2 * t$95$2), $MachinePrecision], N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3e6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 17.7

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 17.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 78.6%

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

   11. Applied rewrites 81.5%

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

   if 2.3e6 < t < 1.7499999999999999e187

   1. Initial program 71.8%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.2%

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

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 91.2%

       \[\leadsto \frac{\frac{-\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{-\ell}{k}}{{t}^{1.5}}} \]

   if 1.7499999999999999e187 < t

   1. Initial program 70.8%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 63.4%

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

   9. Applied rewrites 63.4%

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

   11. Applied rewrites 89.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2300000:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \frac{\frac{\ell}{k}}{{t}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t}\right)}^{2}}{\left(k \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2300000:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t\_m}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2300000.0)
    (/ 2.0 (* (/ (* (* (/ k l) k) t_m) l) (* (tan k) (sin k))))
    (if (<= t_m 9.5e+205)
      (pow (* (/ (pow t_m 1.5) l) k) -2.0)
      (/ (pow (/ l t_m) 2.0) (* (* k t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2300000.0) {
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)));
	} else if (t_m <= 9.5e+205) {
		tmp = pow(((pow(t_m, 1.5) / l) * k), -2.0);
	} else {
		tmp = pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2300000.0d0) then
        tmp = 2.0d0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)))
    else if (t_m <= 9.5d+205) then
        tmp = (((t_m ** 1.5d0) / l) * k) ** (-2.0d0)
    else
        tmp = ((l / t_m) ** 2.0d0) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2300000.0) {
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (Math.tan(k) * Math.sin(k)));
	} else if (t_m <= 9.5e+205) {
		tmp = Math.pow(((Math.pow(t_m, 1.5) / l) * k), -2.0);
	} else {
		tmp = Math.pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2300000.0:
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (math.tan(k) * math.sin(k)))
	elif t_m <= 9.5e+205:
		tmp = math.pow(((math.pow(t_m, 1.5) / l) * k), -2.0)
	else:
		tmp = math.pow((l / t_m), 2.0) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2300000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * k) * t_m) / l) * Float64(tan(k) * sin(k))));
	elseif (t_m <= 9.5e+205)
		tmp = Float64(Float64((t_m ^ 1.5) / l) * k) ^ -2.0;
	else
		tmp = Float64((Float64(l / t_m) ^ 2.0) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2300000.0)
		tmp = 2.0 / (((((k / l) * k) * t_m) / l) * (tan(k) * sin(k)));
	elseif (t_m <= 9.5e+205)
		tmp = (((t_m ^ 1.5) / l) * k) ^ -2.0;
	else
		tmp = ((l / t_m) ^ 2.0) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2300000.0], N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+205], N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3e6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval 17.7

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}} \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 17.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5} \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 78.6%

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

   11. Applied rewrites 81.5%

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

   if 2.3e6 < t < 9.4999999999999997e205

   1. Initial program 70.0%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 69.5

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

   5. Applied rewrites 69.5%

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

   7. Applied rewrites 69.5%

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

   9. Applied rewrites 72.2%

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

   11. Applied rewrites 88.0%

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

   if 9.4999999999999997e205 < t

   1. Initial program 74.2%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 65.5

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

   5. Applied rewrites 65.5%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 65.5%

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

   11. Applied rewrites 91.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2300000:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t}\right)}^{2}}{\left(k \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t\_m}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1)
    (/ (* (/ l t_m) (/ (/ (/ l k) k) t_m)) t_m)
    (if (<= t_m 9.5e+205)
      (pow (* (/ (pow t_m 1.5) l) k) -2.0)
      (/ (pow (/ l t_m) 2.0) (* (* k t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else if (t_m <= 9.5e+205) {
		tmp = pow(((pow(t_m, 1.5) / l) * k), -2.0);
	} else {
		tmp = pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.1d0) then
        tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
    else if (t_m <= 9.5d+205) then
        tmp = (((t_m ** 1.5d0) / l) * k) ** (-2.0d0)
    else
        tmp = ((l / t_m) ** 2.0d0) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else if (t_m <= 9.5e+205) {
		tmp = Math.pow(((Math.pow(t_m, 1.5) / l) * k), -2.0);
	} else {
		tmp = Math.pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.1:
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
	elif t_m <= 9.5e+205:
		tmp = math.pow(((math.pow(t_m, 1.5) / l) * k), -2.0)
	else:
		tmp = math.pow((l / t_m), 2.0) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1)
		tmp = Float64(Float64(Float64(l / t_m) * Float64(Float64(Float64(l / k) / k) / t_m)) / t_m);
	elseif (t_m <= 9.5e+205)
		tmp = Float64(Float64((t_m ^ 1.5) / l) * k) ^ -2.0;
	else
		tmp = Float64((Float64(l / t_m) ^ 2.0) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.1)
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	elseif (t_m <= 9.5e+205)
		tmp = (((t_m ^ 1.5) / l) * k) ^ -2.0;
	else
		tmp = ((l / t_m) ^ 2.0) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+205], N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.10000000000000009

   1. Initial program 51.1%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 52.4%

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

   9. Applied rewrites 56.9%

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

   11. Applied rewrites 69.3%

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

   if 2.10000000000000009 < t < 9.4999999999999997e205

   1. Initial program 71.8%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 88.7%

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

   if 9.4999999999999997e205 < t

   1. Initial program 74.2%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 65.5

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

   5. Applied rewrites 65.5%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 65.5%

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

   11. Applied rewrites 91.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{t}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+205}:\\ \;\;\;\;{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t}\right)}^{2}}{\left(k \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{k \cdot t\_m}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e+84)
    (/ (* (/ l t_m) (/ (/ (/ l k) k) t_m)) t_m)
    (/ (/ (pow (/ l t_m) 2.0) (* k t_m)) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e+84) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else {
		tmp = (pow((l / t_m), 2.0) / (k * t_m)) / k;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.2d+84) then
        tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
    else
        tmp = (((l / t_m) ** 2.0d0) / (k * t_m)) / k
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e+84) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else {
		tmp = (Math.pow((l / t_m), 2.0) / (k * t_m)) / k;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.2e+84:
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
	else:
		tmp = (math.pow((l / t_m), 2.0) / (k * t_m)) / k
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e+84)
		tmp = Float64(Float64(Float64(l / t_m) * Float64(Float64(Float64(l / k) / k) / t_m)) / t_m);
	else
		tmp = Float64(Float64((Float64(l / t_m) ^ 2.0) / Float64(k * t_m)) / k);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.2e+84)
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	else
		tmp = (((l / t_m) ^ 2.0) / (k * t_m)) / k;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e+84], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2000000000000002e84

   1. Initial program 54.5%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 56.9%

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

   9. Applied rewrites 60.2%

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

   11. Applied rewrites 72.4%

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

   if 5.2000000000000002e84 < t

   1. Initial program 70.0%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 59.5%

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

   9. Applied rewrites 64.9%

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

   11. Applied rewrites 87.5%

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

Alternative 10: 71.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.02 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.02e+93)
    (/ (* (/ l t_m) (/ (/ (/ l k) k) t_m)) t_m)
    (/ (pow (/ l t_m) 2.0) (* (* k t_m) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.02e+93) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else {
		tmp = pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.02d+93) then
        tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
    else
        tmp = ((l / t_m) ** 2.0d0) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.02e+93) {
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	} else {
		tmp = Math.pow((l / t_m), 2.0) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.02e+93:
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m
	else:
		tmp = math.pow((l / t_m), 2.0) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.02e+93)
		tmp = Float64(Float64(Float64(l / t_m) * Float64(Float64(Float64(l / k) / k) / t_m)) / t_m);
	else
		tmp = Float64((Float64(l / t_m) ^ 2.0) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.02e+93)
		tmp = ((l / t_m) * (((l / k) / k) / t_m)) / t_m;
	else
		tmp = ((l / t_m) ^ 2.0) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.02e+93], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.01999999999999998e93

   1. Initial program 54.5%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 56.9%

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

   9. Applied rewrites 60.1%

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

   11. Applied rewrites 72.2%

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

   if 2.01999999999999998e93 < t

   1. Initial program 70.9%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   9. Applied rewrites 65.5%

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

   11. Applied rewrites 86.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.02 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t}\right)}^{2}}{\left(k \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 57.4

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

5. Applied rewrites 57.4%

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

7. Applied rewrites 57.4%

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

9. Applied rewrites 61.0%

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

11. Applied rewrites 71.8%

    \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{\color{blue}{t}} \]
12. Add Preprocessing

Alternative 12: 66.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\ell}{k \cdot t\_m} \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 4.2e-131)
    (* (/ l (* k t_m)) (/ (/ l (* t_m t_m)) k))
    (/ (* (/ l t_m) (/ l t_m)) (* t_m (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.2e-131) {
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.2d-131) then
        tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k)
    else
        tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.2e-131) {
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4.2e-131:
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k)
	else:
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.2e-131)
		tmp = Float64(Float64(l / Float64(k * t_m)) * Float64(Float64(l / Float64(t_m * t_m)) / k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(t_m * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4.2e-131)
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k);
	else
		tmp = ((l / t_m) * (l / t_m)) / (t_m * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.2e-131], N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999994e-131

   1. Initial program 58.4%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.8

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

   5. Applied rewrites 57.8%

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

   7. Applied rewrites 57.8%

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

   9. Applied rewrites 61.6%

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

   11. Applied rewrites 67.7%

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

   if 4.19999999999999994e-131 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 56.2%

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

   9. Applied rewrites 59.3%

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

   11. Applied rewrites 65.1%

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

Alternative 13: 66.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-136}:\\ \;\;\;\;\frac{\ell}{k \cdot t\_m} \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.65e-136)
    (* (/ l (* k t_m)) (/ (/ l (* t_m t_m)) k))
    (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.65e-136) {
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.65d-136) then
        tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k)
    else
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.65e-136) {
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.65e-136:
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k)
	else:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.65e-136)
		tmp = Float64(Float64(l / Float64(k * t_m)) * Float64(Float64(l / Float64(t_m * t_m)) / k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.65e-136)
		tmp = (l / (k * t_m)) * ((l / (t_m * t_m)) / k);
	else
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.65e-136], N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.65000000000000009e-136

   1. Initial program 58.2%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

   9. Applied rewrites 61.4%

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

   11. Applied rewrites 67.6%

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

   if 1.65000000000000009e-136 < k

   1. Initial program 54.8%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 56.9%

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

   9. Applied rewrites 59.9%

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

   11. Applied rewrites 62.8%

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

Alternative 14: 62.6% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 57.4

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

5. Applied rewrites 57.4%

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

7. Applied rewrites 57.4%

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

9. Applied rewrites 61.0%

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

11. Applied rewrites 65.8%

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

Alternative 15: 58.9% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 57.4

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

5. Applied rewrites 57.4%

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

7. Applied rewrites 57.4%

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

9. Applied rewrites 61.0%

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

11. Applied rewrites 61.3%

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

Alternative 16: 53.6% accurate, 12.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 57.4

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

5. Applied rewrites 57.4%

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

7. Applied rewrites 57.4%

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

9. Applied rewrites 61.0%

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

11. Applied rewrites 59.3%

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

Reproduce

herbie shell --seed 2024324 
(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))))