Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 95.3%

Time: 29.3s

Alternatives: 11

Speedup: 3.8×

• 39.3% 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: 35.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \left(\frac{\sin k\_m}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \left(2 \cdot \frac{\frac{\cos k\_m}{t\_m}}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.9e-26)
    (/ 2.0 (pow (* k_m (* (/ (sin k_m) l) (sqrt (/ t_m (cos k_m))))) 2.0))
    (*
     (* (/ l k_m) (/ l k_m))
     (* 2.0 (/ (/ (cos k_m) t_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-26) {
		tmp = 2.0 / pow((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((cos(k_m) / t_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.9d-26) then
        tmp = 2.0d0 / ((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))) ** 2.0d0)
    else
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 * ((cos(k_m) / t_m) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 1.9e-26) {
		tmp = 2.0 / Math.pow((k_m * ((Math.sin(k_m) / l) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.9e-26:
		tmp = 2.0 / math.pow((k_m * ((math.sin(k_m) / l) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	else:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((math.cos(k_m) / t_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-26)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(sin(k_m) / l) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 * Float64(Float64(cos(k_m) / t_m) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.9e-26)
		tmp = 2.0 / ((k_m * ((sin(k_m) / l) * sqrt((t_m / cos(k_m))))) ^ 2.0);
	else
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((cos(k_m) / t_m) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-26], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.90000000000000007e-26

   1. Initial program 39.3%

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

      1. add-sqr-sqrt 19.6%

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

      2. pow2 19.6%

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

   4. Applied egg-rr 32.8%

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

   5. Taylor expanded in k around inf 49.4%

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

      1. associate-/l* 50.1%

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

      2. associate-*l* 49.5%

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

   7. Simplified 49.5%

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

   if 1.90000000000000007e-26 < k

   1. Initial program 30.7%

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

      1. *-commutative 30.7%

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

      2. associate-/r* 30.7%

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

   3. Simplified 36.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 36.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 43.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. pow2 43.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 43.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. *-commutative 68.5%

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

      3. rem-square-sqrt 68.4%

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

      4. unpow2 68.4%

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

      5. associate-*r* 68.4%

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

      6. times-frac 69.8%

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

      7. unpow2 69.8%

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

      8. unpow2 69.8%

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

      9. times-frac 91.5%

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

      10. unpow2 91.5%

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

      11. *-commutative 91.5%

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

      12. unpow2 91.5%

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

      13. rem-square-sqrt 91.6%

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

      14. associate-/l* 91.6%

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

      15. associate-/r* 91.7%

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

   9. Simplified 91.7%

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

       1. unpow2 91.7%

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

   11. Applied egg-rr 91.7%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-172}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \left(2 \cdot \frac{\frac{\cos k\_m}{t\_m}}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 4e-172)
    (pow (* (/ l k_m) (/ (sqrt 2.0) (* k_m (sqrt t_m)))) 2.0)
    (*
     (* (/ l k_m) (/ l k_m))
     (* 2.0 (/ (/ (cos k_m) t_m) (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 4e-172) {
		tmp = pow(((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))), 2.0);
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((cos(k_m) / t_m) / pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 4d-172) then
        tmp = ((l / k_m) * (sqrt(2.0d0) / (k_m * sqrt(t_m)))) ** 2.0d0
    else
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 * ((cos(k_m) / t_m) / (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (l <= 4e-172) {
		tmp = Math.pow(((l / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 4e-172:
		tmp = math.pow(((l / k_m) * (math.sqrt(2.0) / (k_m * math.sqrt(t_m)))), 2.0)
	else:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((math.cos(k_m) / t_m) / math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 4e-172)
		tmp = Float64(Float64(l / k_m) * Float64(sqrt(2.0) / Float64(k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 * Float64(Float64(cos(k_m) / t_m) / (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 4e-172)
		tmp = ((l / k_m) * (sqrt(2.0) / (k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = ((l / k_m) * (l / k_m)) * (2.0 * ((cos(k_m) / t_m) / (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[l, 4e-172], N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.0000000000000002e-172

   1. Initial program 33.6%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in t around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. associate-*r* 72.2%

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

      3. times-frac 73.4%

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

      4. *-commutative 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in k around 0 65.7%

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

      1. add-sqr-sqrt 48.3%

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

      2. pow2 48.3%

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

      3. pow2 48.3%

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

      4. sqrt-prod 41.3%

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

      5. sqrt-div 28.0%

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

      6. sqrt-prod 28.0%

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

      7. sqrt-pow1 28.0%

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

      8. metadata-eval 28.0%

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

      9. pow1 28.0%

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

      10. sqrt-div 28.0%

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

      11. sqrt-prod 7.8%

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

      12. add-sqr-sqrt 35.2%

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

      13. sqrt-pow1 35.9%

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

      14. metadata-eval 35.9%

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

      15. pow1 35.9%

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

   10. Applied egg-rr 35.9%

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

   if 4.0000000000000002e-172 < l

   1. Initial program 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-/r* 43.8%

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

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 49.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 58.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. pow2 58.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 58.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around inf 79.7%

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

      1. associate-*r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. rem-square-sqrt 79.5%

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

      4. unpow2 79.5%

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

      5. associate-*r* 79.5%

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

      6. times-frac 80.8%

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

      7. unpow2 80.8%

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

      8. unpow2 80.8%

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

      9. times-frac 96.0%

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

      10. unpow2 96.0%

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

      11. *-commutative 96.0%

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

      12. unpow2 96.0%

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

      13. rem-square-sqrt 96.1%

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

      14. associate-/l* 96.1%

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

      15. associate-/r* 96.2%

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

   9. Simplified 96.2%

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

       1. unpow2 96.2%

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

   11. Applied egg-rr 96.2%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-172}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * Math.pow(((Math.sqrt(2.0) / k_m) * ((l / k_m) / Math.sqrt(t_m))), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Simplified 45.6%

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

5. Taylor expanded in t around 0 74.6%

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

   1. associate-*r/ 74.6%

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

   2. associate-*r* 74.6%

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

   3. times-frac 75.8%

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

   4. *-commutative 75.8%

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

7. Simplified 75.8%

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

8. Taylor expanded in k around 0 65.4%

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

   1. add-sqr-sqrt 46.5%

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

   2. pow2 46.5%

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

   3. pow2 46.5%

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

   4. sqrt-prod 41.2%

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

   5. sqrt-div 29.8%

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

   6. sqrt-prod 29.8%

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

   7. sqrt-pow1 29.8%

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

   8. metadata-eval 29.8%

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

   9. pow1 29.8%

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

   10. sqrt-div 29.8%

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

   11. sqrt-prod 16.5%

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

   12. add-sqr-sqrt 35.1%

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

   13. sqrt-pow1 35.6%

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

   14. metadata-eval 35.6%

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

   15. pow1 35.6%

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

10. Applied egg-rr 35.6%

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

    1. associate-*l/ 35.6%

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

    2. times-frac 35.6%

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

12. Simplified 35.6%

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

13. Final simplification 35.6%

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

Alternative 4: 76.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * Math.pow(((l / k_m) * (Math.sqrt(2.0) / (k_m * Math.sqrt(t_m)))), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Simplified 45.6%

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

5. Taylor expanded in t around 0 74.6%

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

   1. associate-*r/ 74.6%

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

   2. associate-*r* 74.6%

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

   3. times-frac 75.8%

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

   4. *-commutative 75.8%

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

7. Simplified 75.8%

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

8. Taylor expanded in k around 0 65.4%

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

   1. add-sqr-sqrt 46.5%

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

   2. pow2 46.5%

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

   3. pow2 46.5%

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

   4. sqrt-prod 41.2%

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

   5. sqrt-div 29.8%

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

   6. sqrt-prod 29.8%

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

   7. sqrt-pow1 29.8%

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

   8. metadata-eval 29.8%

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

   9. pow1 29.8%

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

   10. sqrt-div 29.8%

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

   11. sqrt-prod 16.5%

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

   12. add-sqr-sqrt 35.1%

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

   13. sqrt-pow1 35.6%

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

   14. metadata-eval 35.6%

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

   15. pow1 35.6%

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

10. Applied egg-rr 35.6%

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

11. Final simplification 35.6%

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

Alternative 5: 74.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k_m, 2.0) / l)), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

   1. add-sqr-sqrt 17.9%

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

   2. pow2 17.9%

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

4. Applied egg-rr 27.9%

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

5. Taylor expanded in k around 0 35.1%

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

6. Final simplification 35.1%

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

Alternative 6: 73.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-186}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{2}}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \left(2 \cdot \frac{\frac{1}{{k\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-186)
    (* (* (/ l k_m) (/ l k_m)) (/ 2.0 (* t_m (pow k_m 2.0))))
    (if (<= t_m 5e+27)
      (/ 2.0 (pow (* (/ k_m t_m) (* k_m (/ (pow t_m 1.5) l))) 2.0))
      (* (pow (/ l k_m) 2.0) (* 2.0 (/ (/ 1.0 (pow k_m 2.0)) t_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5.2e-186) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t_m * pow(k_m, 2.0)));
	} else if (t_m <= 5e+27) {
		tmp = 2.0 / pow(((k_m / t_m) * (k_m * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = pow((l / k_m), 2.0) * (2.0 * ((1.0 / pow(k_m, 2.0)) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5.2d-186) then
        tmp = ((l / k_m) * (l / k_m)) * (2.0d0 / (t_m * (k_m ** 2.0d0)))
    else if (t_m <= 5d+27) then
        tmp = 2.0d0 / (((k_m / t_m) * (k_m * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    else
        tmp = ((l / k_m) ** 2.0d0) * (2.0d0 * ((1.0d0 / (k_m ** 2.0d0)) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 5.2e-186) {
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t_m * Math.pow(k_m, 2.0)));
	} else if (t_m <= 5e+27) {
		tmp = 2.0 / Math.pow(((k_m / t_m) * (k_m * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = Math.pow((l / k_m), 2.0) * (2.0 * ((1.0 / Math.pow(k_m, 2.0)) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 5.2e-186:
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t_m * math.pow(k_m, 2.0)))
	elif t_m <= 5e+27:
		tmp = 2.0 / math.pow(((k_m / t_m) * (k_m * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = math.pow((l / k_m), 2.0) * (2.0 * ((1.0 / math.pow(k_m, 2.0)) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 5.2e-186)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(2.0 / Float64(t_m * (k_m ^ 2.0))));
	elseif (t_m <= 5e+27)
		tmp = Float64(2.0 / (Float64(Float64(k_m / t_m) * Float64(k_m * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	else
		tmp = Float64((Float64(l / k_m) ^ 2.0) * Float64(2.0 * Float64(Float64(1.0 / (k_m ^ 2.0)) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 5.2e-186)
		tmp = ((l / k_m) * (l / k_m)) * (2.0 / (t_m * (k_m ^ 2.0)));
	elseif (t_m <= 5e+27)
		tmp = 2.0 / (((k_m / t_m) * (k_m * ((t_m ^ 1.5) / l))) ^ 2.0);
	else
		tmp = ((l / k_m) ^ 2.0) * (2.0 * ((1.0 / (k_m ^ 2.0)) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-186], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+27], N[(2.0 / N[Power[N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.19999999999999986e-186

   1. Initial program 36.9%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. associate-*r* 73.2%

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

      3. times-frac 75.0%

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

      4. *-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in k around 0 64.1%

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

      1. pow2 64.1%

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

      2. add-sqr-sqrt 64.1%

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

      3. sqrt-div 64.1%

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

      4. sqrt-prod 24.6%

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

      5. add-sqr-sqrt 42.5%

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

      6. sqrt-pow1 50.9%

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

      7. metadata-eval 50.9%

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

      8. pow1 50.9%

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

      9. sqrt-div 50.9%

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

      10. sqrt-prod 21.1%

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

      11. add-sqr-sqrt 51.8%

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

      12. sqrt-pow1 71.0%

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

      13. metadata-eval 71.0%

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

      14. pow1 71.0%

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

   10. Applied egg-rr 71.0%

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

   if 5.19999999999999986e-186 < t < 4.99999999999999979e27

   1. Initial program 42.6%

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

      1. add-sqr-sqrt 32.9%

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

      2. pow2 32.9%

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

   4. Applied egg-rr 72.4%

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

   5. Taylor expanded in k around 0 74.4%

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

   if 4.99999999999999979e27 < t

   1. Initial program 31.7%

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

      1. *-commutative 31.7%

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

      2. associate-/r* 31.7%

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

   3. Simplified 42.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 42.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. times-frac 46.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. pow2 46.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 46.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around inf 82.1%

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

      1. associate-*r/ 82.1%

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

      2. *-commutative 82.1%

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

      3. rem-square-sqrt 82.0%

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

      4. unpow2 82.0%

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

      5. associate-*r* 82.0%

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

      6. times-frac 87.9%

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

      7. unpow2 87.9%

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

      8. unpow2 87.9%

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

      9. times-frac 95.8%

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

      10. unpow2 95.8%

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

      11. *-commutative 95.8%

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

      12. unpow2 95.8%

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

      13. rem-square-sqrt 95.9%

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

      14. associate-/l* 95.9%

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

      15. associate-/r* 96.0%

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

   9. Simplified 96.0%

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

   10. Taylor expanded in k around 0 82.1%

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

       1. associate-/r* 82.1%

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

   12. Simplified 82.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-186}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \left(2 \cdot \frac{\frac{1}{{k}^{2}}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (pow (/ l k_m) 2.0) (* 2.0 (/ (/ 1.0 (pow k_m 2.0)) t_m)))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (Math.pow((l / k_m), 2.0) * (2.0 * ((1.0 / Math.pow(k_m, 2.0)) / t_m)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/r* 36.9%

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

3. Simplified 45.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow3 45.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. times-frac 56.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. pow2 56.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Applied egg-rr 56.8%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around inf 74.6%

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

   1. associate-*r/ 74.6%

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

   2. *-commutative 74.6%

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

   3. rem-square-sqrt 74.5%

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

   4. unpow2 74.5%

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

   5. associate-*r* 74.5%

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

   6. times-frac 75.7%

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

   7. unpow2 75.7%

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

   8. unpow2 75.7%

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

   9. times-frac 91.4%

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

   10. unpow2 91.4%

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

   11. *-commutative 91.4%

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

   12. unpow2 91.4%

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

   13. rem-square-sqrt 91.6%

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

   14. associate-/l* 91.6%

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

   15. associate-/r* 91.6%

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

9. Simplified 91.6%

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

10. Taylor expanded in k around 0 72.2%

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

    1. associate-/r* 72.2%

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

12. Simplified 72.2%

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

13. Final simplification 72.2%

    \[\leadsto {\left(\frac{\ell}{k}\right)}^{2} \cdot \left(2 \cdot \frac{\frac{1}{{k}^{2}}}{t}\right) \]
14. Add Preprocessing

Alternative 8: 72.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((2.0 * Math.pow((l / k_m), 2.0)) / (t_m * Math.pow(k_m, 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(2.0 * N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Simplified 45.6%

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

5. Taylor expanded in t around 0 74.6%

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

   1. associate-*r/ 74.6%

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

   2. associate-*r* 74.6%

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

   3. times-frac 75.8%

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

   4. *-commutative 75.8%

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

7. Simplified 75.8%

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

8. Taylor expanded in k around 0 65.4%

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

   1. associate-*l/ 65.4%

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

   2. pow2 65.4%

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

   3. add-sqr-sqrt 65.4%

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

   4. pow2 65.4%

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

   5. sqrt-div 65.4%

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

   6. sqrt-prod 30.3%

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

   7. add-sqr-sqrt 72.2%

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

   8. sqrt-pow1 72.2%

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

   9. metadata-eval 72.2%

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

   10. pow1 72.2%

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

   11. *-commutative 72.2%

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

10. Applied egg-rr 72.2%

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

11. Final simplification 72.2%

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

Alternative 9: 72.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (((l / k_m) * (l / k_m)) * (2.0 / (t_m * Math.pow(k_m, 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Simplified 45.6%

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

5. Taylor expanded in t around 0 74.6%

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

   1. associate-*r/ 74.6%

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

   2. associate-*r* 74.6%

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

   3. times-frac 75.8%

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

   4. *-commutative 75.8%

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

7. Simplified 75.8%

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

8. Taylor expanded in k around 0 65.4%

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

   1. pow2 65.4%

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

   2. add-sqr-sqrt 65.3%

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

   3. sqrt-div 65.3%

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

   4. sqrt-prod 27.8%

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

   5. add-sqr-sqrt 47.8%

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

   6. sqrt-pow1 52.9%

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

   7. metadata-eval 52.9%

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

   8. pow1 52.9%

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

   9. sqrt-div 52.9%

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

   10. sqrt-prod 22.5%

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

   11. add-sqr-sqrt 52.4%

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

   12. sqrt-pow1 72.2%

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

   13. metadata-eval 72.2%

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

   14. pow1 72.2%

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

10. Applied egg-rr 72.2%

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

11. Final simplification 72.2%

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

Alternative 10: 62.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{{k\_m}^{-4}}{t\_m}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (* 2.0 (/ (pow k_m -4.0) t_m)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (2.0 * (pow(k_m, -4.0) / t_m)));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * (2.0d0 * ((k_m ** (-4.0d0)) / t_m)))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * (2.0 * (Math.pow(k_m, -4.0) / t_m)));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (2.0 * (math.pow(k_m, -4.0) / t_m)))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 * Float64((k_m ^ -4.0) / t_m))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (2.0 * ((k_m ^ -4.0) / t_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Power[k$95$m, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Simplified 45.6%

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

5. Taylor expanded in k around 0 61.8%

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

   1. div-inv 61.8%

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

   2. *-commutative 61.8%

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

7. Applied egg-rr 61.8%

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

   1. associate-*r/ 61.8%

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

   2. metadata-eval 61.8%

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

   3. associate-/r* 61.9%

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

9. Simplified 61.9%

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

    1. div-inv 61.9%

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

    2. pow-flip 61.9%

       \[\leadsto \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

    3. metadata-eval 61.9%

       \[\leadsto \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right) \]

11. Applied egg-rr 61.9%

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

    1. associate-*l/ 61.9%

       \[\leadsto \color{blue}{\frac{2 \cdot {k}^{-4}}{t}} \cdot \left(\ell \cdot \ell\right) \]

    2. associate-/l* 61.9%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{{k}^{-4}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

13. Simplified 61.9%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{{k}^{-4}}{t}\right)} \cdot \left(\ell \cdot \ell\right) \]

14. Final simplification 61.9%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{{k}^{-4}}{t}\right) \]
15. Add Preprocessing

Alternative 11: 62.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right) \cdot \left(\ell \cdot \ell\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* (/ 2.0 t_m) (pow k_m -4.0)) (* l l))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (((2.0 / t_m) * pow(k_m, -4.0)) * (l * l));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (((2.0d0 / t_m) * (k_m ** (-4.0d0))) * (l * l))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (((2.0 / t_m) * Math.pow(k_m, -4.0)) * (l * l));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (((2.0 / t_m) * math.pow(k_m, -4.0)) * (l * l))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(Float64(2.0 / t_m) * (k_m ^ -4.0)) * Float64(l * l)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (((2.0 / t_m) * (k_m ^ -4.0)) * (l * l));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.9%

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

3. Simplified 45.6%

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

5. Taylor expanded in k around 0 61.8%

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

   1. div-inv 61.8%

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

   2. *-commutative 61.8%

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

7. Applied egg-rr 61.8%

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

   1. associate-*r/ 61.8%

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

   2. metadata-eval 61.8%

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

   3. associate-/r* 61.9%

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

9. Simplified 61.9%

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

    1. div-inv 61.9%

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

    2. pow-flip 61.9%

       \[\leadsto \left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

    3. metadata-eval 61.9%

       \[\leadsto \left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right) \]

11. Applied egg-rr 61.9%

    \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right)} \cdot \left(\ell \cdot \ell\right) \]

12. Final simplification 61.9%

    \[\leadsto \left(\frac{2}{t} \cdot {k}^{-4}\right) \cdot \left(\ell \cdot \ell\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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))))