Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.7% → 94.9%

Time: 15.5s

Alternatives: 6

Speedup: 3.8×

• 39.7% 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: 34.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{k\_m}{\ell}\right) \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right)}^{2}}}{\frac{t\_m}{\cos k\_m}}\\ \end{array} \end{array} \]

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 1.7e-35)
    (/ 2.0 (pow (* (* k_m (/ k_m l)) (sqrt t_m)) 2.0))
    (/ (/ 2.0 (pow (* k_m (/ (sin k_m) l)) 2.0)) (/ t_m (cos k_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 (k_m <= 1.7e-35) {
		tmp = 2.0 / pow(((k_m * (k_m / l)) * sqrt(t_m)), 2.0);
	} else {
		tmp = (2.0 / pow((k_m * (sin(k_m) / l)), 2.0)) / (t_m / cos(k_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 (k_m <= 1.7d-35) then
        tmp = 2.0d0 / (((k_m * (k_m / l)) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = (2.0d0 / ((k_m * (sin(k_m) / l)) ** 2.0d0)) / (t_m / cos(k_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 (k_m <= 1.7e-35) {
		tmp = 2.0 / Math.pow(((k_m * (k_m / l)) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = (2.0 / Math.pow((k_m * (Math.sin(k_m) / l)), 2.0)) / (t_m / Math.cos(k_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 k_m <= 1.7e-35:
		tmp = 2.0 / math.pow(((k_m * (k_m / l)) * math.sqrt(t_m)), 2.0)
	else:
		tmp = (2.0 / math.pow((k_m * (math.sin(k_m) / l)), 2.0)) / (t_m / math.cos(k_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 (k_m <= 1.7e-35)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(k_m / l)) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / (Float64(k_m * Float64(sin(k_m) / l)) ^ 2.0)) / Float64(t_m / cos(k_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 (k_m <= 1.7e-35)
		tmp = 2.0 / (((k_m * (k_m / l)) * sqrt(t_m)) ^ 2.0);
	else
		tmp = (2.0 / ((k_m * (sin(k_m) / l)) ^ 2.0)) / (t_m / cos(k_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[k$95$m, 1.7e-35], N[(2.0 / N[Power[N[(N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7000000000000001e-35

   1. Initial program 41.0%

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

      1. *-commutative 41.0%

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

      2. associate-/r* 41.0%

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

   3. Simplified 46.8%

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

      1. +-rgt-identity 46.8%

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

      2. associate-/l/ 46.8%

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

      3. div-inv 46.8%

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

      4. +-rgt-identity 46.8%

         \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      5. metadata-eval 46.8%

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

      6. metadata-eval 46.8%

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

      7. associate-+l+ 41.0%

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

      8. sub-neg 41.0%

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

      9. +-commutative 41.0%

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

      10. add-sqr-sqrt 20.8%

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

   6. Applied egg-rr 34.2%

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

      1. associate-*r/ 34.2%

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

      2. metadata-eval 34.2%

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

      3. *-commutative 34.2%

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

      4. associate-*l* 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in k around 0 40.3%

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

       1. unpow2 40.3%

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

       2. add-sqr-sqrt 21.3%

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

       3. times-frac 21.7%

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

   11. Applied egg-rr 21.7%

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

       1. unpow2 21.7%

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

   13. Simplified 21.7%

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

       1. unpow2 21.7%

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

       2. frac-times 21.3%

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

       3. add-sqr-sqrt 40.3%

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

       4. associate-*r/ 41.2%

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

       5. *-commutative 41.2%

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

   15. Applied egg-rr 41.2%

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

   if 1.7000000000000001e-35 < k

   1. Initial program 29.5%

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

      1. *-commutative 29.5%

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

      2. associate-/r* 29.5%

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

   3. Simplified 47.0%

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

      1. +-rgt-identity 47.0%

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

      2. associate-/l/ 47.1%

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

      3. div-inv 47.1%

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

      4. +-rgt-identity 47.1%

         \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      5. metadata-eval 47.1%

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

      6. metadata-eval 47.1%

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

      7. associate-+l+ 29.5%

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

      8. sub-neg 29.5%

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

      9. +-commutative 29.5%

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

      10. add-sqr-sqrt 11.7%

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

   6. Applied egg-rr 27.9%

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

      1. associate-*r/ 27.9%

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

      2. metadata-eval 27.9%

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

      3. *-commutative 27.9%

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

      4. associate-*l* 28.0%

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

   8. Simplified 28.0%

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

   9. Taylor expanded in k around inf 47.0%

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

       1. associate-*l/ 45.3%

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

       2. associate-/l* 47.1%

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

   11. Simplified 47.1%

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

       1. add-sqr-sqrt 47.0%

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

       2. sqrt-div 47.1%

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

       3. sqrt-pow1 36.8%

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

       4. metadata-eval 36.8%

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

       5. pow1 36.8%

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

       6. sqrt-div 36.7%

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

       7. sqrt-pow1 46.9%

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

       8. metadata-eval 46.9%

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

       9. pow1 46.9%

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

   13. Applied egg-rr 46.9%

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

       1. unpow2 46.9%

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

       2. associate-*r/ 45.2%

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

       3. associate-*l/ 46.8%

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

       4. associate-/l* 46.9%

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

   15. Simplified 46.9%

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

       1. unpow2 46.9%

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

       2. associate-/r* 47.0%

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

       3. associate-/r* 46.9%

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

       4. frac-times 43.7%

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

       5. add-sqr-sqrt 95.1%

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

   17. Applied egg-rr 95.1%

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

       1. associate-*l/ 95.1%

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

       2. associate-*r/ 94.9%

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

       3. rem-square-sqrt 95.2%

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

       4. associate-/l/ 95.3%

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

       5. unpow2 95.3%

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

   19. Simplified 95.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot \frac{\sin k}{\ell}\right)}^{2}}}{\frac{t}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.04:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{k\_m}{\ell}\right) \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \sin k\_m\right) \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 0.04)
    (/ 2.0 (pow (* (* k_m (/ k_m l)) (sqrt t_m)) 2.0))
    (/ 2.0 (pow (* (* k_m (sin k_m)) (/ (sqrt t_m) 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) {
	double tmp;
	if (k_m <= 0.04) {
		tmp = 2.0 / pow(((k_m * (k_m / l)) * sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / pow(((k_m * sin(k_m)) * (sqrt(t_m) / l)), 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 <= 0.04d0) then
        tmp = 2.0d0 / (((k_m * (k_m / l)) * sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 / (((k_m * sin(k_m)) * (sqrt(t_m) / l)) ** 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 <= 0.04) {
		tmp = 2.0 / Math.pow(((k_m * (k_m / l)) * Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((k_m * Math.sin(k_m)) * (Math.sqrt(t_m) / l)), 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 <= 0.04:
		tmp = 2.0 / math.pow(((k_m * (k_m / l)) * math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 / math.pow(((k_m * math.sin(k_m)) * (math.sqrt(t_m) / l)), 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 <= 0.04)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(k_m / l)) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(k_m * sin(k_m)) * Float64(sqrt(t_m) / l)) ^ 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 <= 0.04)
		tmp = 2.0 / (((k_m * (k_m / l)) * sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 / (((k_m * sin(k_m)) * (sqrt(t_m) / l)) ^ 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, 0.04], N[(2.0 / N[Power[N[(N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0400000000000000008

   1. Initial program 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-/r* 40.2%

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

   3. Simplified 45.9%

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

      1. +-rgt-identity 45.9%

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

      2. associate-/l/ 45.9%

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

      3. div-inv 45.9%

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

      4. +-rgt-identity 45.9%

         \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      5. metadata-eval 45.9%

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

      6. metadata-eval 45.9%

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

      7. associate-+l+ 40.2%

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

      8. sub-neg 40.2%

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

      9. +-commutative 40.2%

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

      10. add-sqr-sqrt 20.4%

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

   6. Applied egg-rr 33.6%

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

      1. associate-*r/ 33.6%

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

      2. metadata-eval 33.6%

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

      3. *-commutative 33.6%

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

      4. associate-*l* 33.6%

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

   8. Simplified 33.6%

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

   9. Taylor expanded in k around 0 40.4%

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

       1. unpow2 40.4%

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

       2. add-sqr-sqrt 20.9%

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

       3. times-frac 21.3%

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

   11. Applied egg-rr 21.3%

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

       1. unpow2 21.3%

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

   13. Simplified 21.3%

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

       1. unpow2 21.3%

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

       2. frac-times 20.9%

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

       3. add-sqr-sqrt 40.4%

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

       4. associate-*r/ 41.4%

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

       5. *-commutative 41.4%

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

   15. Applied egg-rr 41.4%

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

   if 0.0400000000000000008 < k

   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 50.5%

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

      1. +-rgt-identity 50.5%

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

      2. associate-/l/ 50.6%

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

      3. div-inv 50.6%

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

      4. +-rgt-identity 50.6%

         \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      5. metadata-eval 50.6%

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

      6. metadata-eval 50.6%

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

      7. associate-+l+ 31.7%

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

      8. sub-neg 31.7%

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

      9. +-commutative 31.7%

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

      10. add-sqr-sqrt 12.6%

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

   6. Applied egg-rr 29.8%

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

      1. associate-*r/ 29.8%

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

      2. metadata-eval 29.8%

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

      3. *-commutative 29.8%

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

      4. associate-*l* 29.9%

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

   8. Simplified 29.9%

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

   9. Taylor expanded in k around inf 46.8%

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

       1. associate-*l/ 45.0%

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

       2. associate-/l* 46.9%

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

   11. Simplified 46.9%

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

   12. Taylor expanded in k around 0 36.3%

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

       1. associate-*l/ 36.3%

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

       2. *-lft-identity 36.3%

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

   14. Simplified 36.3%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.04:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{k}{\ell}\right) \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sin k\right) \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 2.0× speedup

Math

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

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 (/ 2.0 (pow (* (* k_m (/ k_m l)) (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 * (2.0 / pow(((k_m * (k_m / l)) * 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 * (2.0d0 / (((k_m * (k_m / l)) * 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 * (2.0 / Math.pow(((k_m * (k_m / l)) * 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 * (2.0 / math.pow(((k_m * (k_m / l)) * 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(2.0 / (Float64(Float64(k_m * Float64(k_m / l)) * 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 * (2.0 / (((k_m * (k_m / l)) * 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[(2.0 / N[Power[N[(N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. *-commutative 38.4%

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

   2. associate-/r* 38.4%

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

3. Simplified 46.8%

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

   1. +-rgt-identity 46.8%

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

   2. associate-/l/ 46.9%

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

   3. div-inv 46.9%

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

   4. +-rgt-identity 46.9%

      \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

   5. metadata-eval 46.9%

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

   6. metadata-eval 46.9%

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

   7. associate-+l+ 38.4%

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

   8. sub-neg 38.4%

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

   9. +-commutative 38.4%

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

   10. add-sqr-sqrt 18.8%

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

6. Applied egg-rr 32.8%

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

   1. associate-*r/ 32.8%

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

   2. metadata-eval 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*l* 32.9%

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

8. Simplified 32.9%

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

9. Taylor expanded in k around 0 39.2%

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

    1. unpow2 39.2%

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

    2. add-sqr-sqrt 20.1%

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

    3. times-frac 20.5%

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

11. Applied egg-rr 20.5%

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

    1. unpow2 20.5%

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

13. Simplified 20.5%

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

    1. unpow2 20.5%

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

    2. frac-times 20.1%

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

    3. add-sqr-sqrt 39.2%

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

    4. associate-*r/ 39.9%

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

    5. *-commutative 39.9%

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

15. Applied egg-rr 39.9%

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

16. Final simplification 39.9%

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

Alternative 4: 36.6% accurate, 2.0× speedup

Math

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

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 (sqrt l)) 4.0))))

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 / t_m) / pow((k_m / sqrt(l)), 4.0));
}

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 / t_m) / ((k_m / sqrt(l)) ** 4.0d0))
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 / t_m) / Math.pow((k_m / Math.sqrt(l)), 4.0));
}

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 / t_m) / math.pow((k_m / math.sqrt(l)), 4.0))

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 / t_m) / (Float64(k_m / sqrt(l)) ^ 4.0)))
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 / t_m) / ((k_m / sqrt(l)) ^ 4.0));
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 / t$95$m), $MachinePrecision] / N[Power[N[(k$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. *-commutative 38.4%

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

   2. associate-/r* 38.4%

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

3. Simplified 46.8%

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

   1. +-rgt-identity 46.8%

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

   2. associate-/l/ 46.9%

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

   3. div-inv 46.9%

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

   4. +-rgt-identity 46.9%

      \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

   5. metadata-eval 46.9%

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

   6. metadata-eval 46.9%

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

   7. associate-+l+ 38.4%

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

   8. sub-neg 38.4%

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

   9. +-commutative 38.4%

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

   10. add-sqr-sqrt 18.8%

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

6. Applied egg-rr 32.8%

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

   1. associate-*r/ 32.8%

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

   2. metadata-eval 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*l* 32.9%

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

8. Simplified 32.9%

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

9. Taylor expanded in k around 0 39.2%

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

    1. unpow2 39.2%

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

    2. *-un-lft-identity 39.2%

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

    3. metadata-eval 39.2%

       \[\leadsto \frac{2}{{\left(\frac{k \cdot k}{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \ell} \cdot \sqrt{t}\right)}^{2}} \]

    4. times-frac 39.9%

       \[\leadsto \frac{2}{{\left(\color{blue}{\left(\frac{k}{3 \cdot 0.3333333333333333} \cdot \frac{k}{\ell}\right)} \cdot \sqrt{t}\right)}^{2}} \]

    5. metadata-eval 39.9%

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

11. Applied egg-rr 39.9%

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

    1. *-un-lft-identity 39.9%

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

    2. add-sqr-sqrt 39.9%

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

    3. pow2 39.9%

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

13. Applied egg-rr 37.1%

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

    1. *-lft-identity 37.1%

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

    2. associate-/r* 37.5%

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

15. Simplified 37.5%

    \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{\left(\frac{k}{\sqrt{\ell}}\right)}^{4}}} \]
16. Add Preprocessing

Alternative 5: 36.6% accurate, 2.0× speedup

Math

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

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 (sqrt l)) 4.0)))))

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 / (t_m * pow((k_m / sqrt(l)), 4.0)));
}

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 / (t_m * ((k_m / sqrt(l)) ** 4.0d0)))
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 / (t_m * Math.pow((k_m / Math.sqrt(l)), 4.0)));
}

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 / (t_m * math.pow((k_m / math.sqrt(l)), 4.0)))

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(t_m * (Float64(k_m / sqrt(l)) ^ 4.0))))
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 / (t_m * ((k_m / sqrt(l)) ^ 4.0)));
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[(t$95$m * N[Power[N[(k$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. *-commutative 38.4%

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

   2. associate-/r* 38.4%

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

3. Simplified 46.8%

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

   1. +-rgt-identity 46.8%

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

   2. associate-/l/ 46.9%

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

   3. div-inv 46.9%

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

   4. +-rgt-identity 46.9%

      \[\leadsto 2 \cdot \frac{1}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

   5. metadata-eval 46.9%

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

   6. metadata-eval 46.9%

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

   7. associate-+l+ 38.4%

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

   8. sub-neg 38.4%

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

   9. +-commutative 38.4%

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

   10. add-sqr-sqrt 18.8%

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

6. Applied egg-rr 32.8%

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

   1. associate-*r/ 32.8%

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

   2. metadata-eval 32.8%

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

   3. *-commutative 32.8%

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

   4. associate-*l* 32.9%

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

8. Simplified 32.9%

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

9. Taylor expanded in k around 0 39.2%

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

    1. unpow2 39.2%

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

    2. *-un-lft-identity 39.2%

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

    3. metadata-eval 39.2%

       \[\leadsto \frac{2}{{\left(\frac{k \cdot k}{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \ell} \cdot \sqrt{t}\right)}^{2}} \]

    4. times-frac 39.9%

       \[\leadsto \frac{2}{{\left(\color{blue}{\left(\frac{k}{3 \cdot 0.3333333333333333} \cdot \frac{k}{\ell}\right)} \cdot \sqrt{t}\right)}^{2}} \]

    5. metadata-eval 39.9%

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

11. Applied egg-rr 39.9%

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

    1. unpow-prod-down 37.9%

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

    2. frac-times 37.9%

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

    3. *-un-lft-identity 37.9%

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

    4. add-sqr-sqrt 19.2%

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

    5. frac-times 19.2%

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

    6. unpow2 19.2%

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

    7. pow2 19.2%

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

    8. add-sqr-sqrt 37.1%

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

    9. pow-pow 37.1%

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

    10. metadata-eval 37.1%

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

13. Applied egg-rr 37.1%

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

14. Final simplification 37.1%

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

Alternative 6: 62.2% 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 38.4%

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

3. Simplified 46.4%

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

5. Taylor expanded in k around 0 66.4%

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

   1. *-commutative 66.4%

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

   2. associate-/r* 66.4%

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

7. Simplified 66.4%

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

   1. div-inv 66.4%

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

   2. pow-flip 66.4%

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

   3. metadata-eval 66.4%

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

9. Applied egg-rr 66.4%

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

Reproduce

herbie shell --seed 2024103 
(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))))