Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 98.3%

Time: 19.9s

Alternatives: 12

Speedup: 3.8×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 35.2% 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: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\frac{k_m \cdot \sqrt{\frac{t_m}{\cos k_m}}}{\frac{\ell}{\sin k_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{\frac{k_m}{\frac{\ell}{\sqrt{t_m}}}}\right)}^{2} \cdot \frac{\cos k_m}{{\sin k_m}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.8e-18)
    (/ 2.0 (pow (/ (* k_m (sqrt (/ t_m (cos k_m)))) (/ l (sin k_m))) 2.0))
    (*
     2.0
     (*
      (pow (/ 1.0 (/ k_m (/ l (sqrt t_m)))) 2.0)
      (/ (cos k_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 <= 4.8e-18) {
		tmp = 2.0 / pow(((k_m * sqrt((t_m / cos(k_m)))) / (l / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((1.0 / (k_m / (l / sqrt(t_m)))), 2.0) * (cos(k_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 <= 4.8d-18) then
        tmp = 2.0d0 / (((k_m * sqrt((t_m / cos(k_m)))) / (l / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * (((1.0d0 / (k_m / (l / sqrt(t_m)))) ** 2.0d0) * (cos(k_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 <= 4.8e-18) {
		tmp = 2.0 / Math.pow(((k_m * Math.sqrt((t_m / Math.cos(k_m)))) / (l / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((1.0 / (k_m / (l / Math.sqrt(t_m)))), 2.0) * (Math.cos(k_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 <= 4.8e-18:
		tmp = 2.0 / math.pow(((k_m * math.sqrt((t_m / math.cos(k_m)))) / (l / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((1.0 / (k_m / (l / math.sqrt(t_m)))), 2.0) * (math.cos(k_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 <= 4.8e-18)
		tmp = Float64(2.0 / (Float64(Float64(k_m * sqrt(Float64(t_m / cos(k_m)))) / Float64(l / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(1.0 / Float64(k_m / Float64(l / sqrt(t_m)))) ^ 2.0) * Float64(cos(k_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 <= 4.8e-18)
		tmp = 2.0 / (((k_m * sqrt((t_m / cos(k_m)))) / (l / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 * (((1.0 / (k_m / (l / sqrt(t_m)))) ^ 2.0) * (cos(k_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, 4.8e-18], N[(2.0 / N[Power[N[(N[(k$95$m * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(1.0 / N[(k$95$m / N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k$95$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 < 4.79999999999999988e-18

   1. Initial program 39.4%

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

      1. associate-*l* 39.4%

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

      2. associate-*l/ 39.9%

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

      3. associate--l+ 39.9%

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

   3. Simplified 39.9%

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

      1. add-sqr-sqrt 13.7%

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

      2. pow2 13.7%

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

   5. Applied egg-rr 14.8%

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

      1. associate-/l* 15.3%

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

      2. associate-*r/ 15.4%

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

   7. Simplified 15.4%

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

   8. Taylor expanded in k around inf 36.8%

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

      1. associate-/l* 37.3%

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

   10. Simplified 37.3%

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

       1. associate-*l/ 37.4%

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

   12. Applied egg-rr 37.4%

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

   if 4.79999999999999988e-18 < k

   1. Initial program 34.2%

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

      1. associate-/r* 34.2%

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

      2. *-commutative 34.2%

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

      3. associate-*l* 34.2%

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

      4. associate-*l/ 34.1%

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

      5. +-commutative 34.1%

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

      6. unpow2 34.1%

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

      7. sqr-neg 34.1%

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

      8. distribute-frac-neg 34.1%

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

      9. distribute-frac-neg 34.1%

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

      10. unpow2 34.1%

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

      11. associate--l+ 41.9%

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

      12. metadata-eval 41.9%

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

      13. +-rgt-identity 41.9%

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

      14. unpow2 41.9%

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

      15. distribute-frac-neg 41.9%

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

      16. distribute-frac-neg 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around inf 73.5%

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

      1. expm1-log1p-u 62.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} - 1\right)} \]

      3. times-frac 55.6%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 59.7%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-log1p 71.3%

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

      3. times-frac 73.5%

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

      4. associate-*r* 73.5%

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

      5. times-frac 73.5%

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

      6. *-commutative 73.5%

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

   8. Simplified 73.5%

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

      1. expm1-log1p-u 64.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      2. expm1-udef 57.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)} - 1\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      3. pow2 57.9%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \ell}}{t \cdot {k}^{2}}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      4. add-sqr-sqrt 56.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{\ell \cdot \ell}{t \cdot {k}^{2}}} \cdot \sqrt{\frac{\ell \cdot \ell}{t \cdot {k}^{2}}}}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      5. pow2 56.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{\ell \cdot \ell}{t \cdot {k}^{2}}}\right)}^{2}}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      6. sqrt-div 29.2%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{\sqrt{\ell \cdot \ell}}{\sqrt{t \cdot {k}^{2}}}\right)}}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      7. sqrt-prod 12.5%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{t \cdot {k}^{2}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      8. add-sqr-sqrt 33.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{\ell}}{\sqrt{t \cdot {k}^{2}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      9. *-commutative 33.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\sqrt{\color{blue}{{k}^{2} \cdot t}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      10. sqrt-prod 33.7%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{t}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      11. unpow2 33.7%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      12. sqrt-prod 37.0%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      13. add-sqr-sqrt 37.0%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{k} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   10. Applied egg-rr 37.0%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]
   11. Step-by-step derivation

       1. expm1-def 46.3%

          \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}}\right)}^{2}\right)\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

       2. expm1-log1p 47.3%

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

   12. Simplified 47.3%

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

       1. clear-num 47.2%

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

       2. inv-pow 47.2%

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

   14. Applied egg-rr 47.2%

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

       1. unpow-1 47.2%

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

       2. associate-/l* 47.2%

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

   16. Simplified 47.2%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{\frac{t}{\cos k}}}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{\frac{k}{\frac{\ell}{\sqrt{t}}}}\right)}^{2} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{k_m \cdot \sqrt{\frac{t_m}{\cos k_m}}}{\frac{\ell}{\sin k_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k_m}{{\sin k_m}^{2}} \cdot {\left(\frac{\ell}{k_m \cdot \sqrt{t_m}}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.7e-17)
    (/ 2.0 (pow (/ (* k_m (sqrt (/ t_m (cos k_m)))) (/ l (sin k_m))) 2.0))
    (*
     2.0
     (*
      (/ (cos k_m) (pow (sin k_m) 2.0))
      (pow (/ 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) {
	double tmp;
	if (k_m <= 3.7e-17) {
		tmp = 2.0 / pow(((k_m * sqrt((t_m / cos(k_m)))) / (l / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(sin(k_m), 2.0)) * pow((l / (k_m * sqrt(t_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 <= 3.7d-17) then
        tmp = 2.0d0 / (((k_m * sqrt((t_m / cos(k_m)))) / (l / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) / (sin(k_m) ** 2.0d0)) * ((l / (k_m * sqrt(t_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 <= 3.7e-17) {
		tmp = 2.0 / Math.pow(((k_m * Math.sqrt((t_m / Math.cos(k_m)))) / (l / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) * Math.pow((l / (k_m * Math.sqrt(t_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 <= 3.7e-17:
		tmp = 2.0 / math.pow(((k_m * math.sqrt((t_m / math.cos(k_m)))) / (l / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) * math.pow((l / (k_m * math.sqrt(t_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 <= 3.7e-17)
		tmp = Float64(2.0 / (Float64(Float64(k_m * sqrt(Float64(t_m / cos(k_m)))) / Float64(l / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) * (Float64(l / Float64(k_m * sqrt(t_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 <= 3.7e-17)
		tmp = 2.0 / (((k_m * sqrt((t_m / cos(k_m)))) / (l / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) / (sin(k_m) ^ 2.0)) * ((l / (k_m * sqrt(t_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, 3.7e-17], N[(2.0 / N[Power[N[(N[(k$95$m * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.6999999999999997e-17

   1. Initial program 39.4%

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

      1. associate-*l* 39.4%

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

      2. associate-*l/ 39.9%

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

      3. associate--l+ 39.9%

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

   3. Simplified 39.9%

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

      1. add-sqr-sqrt 13.7%

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

      2. pow2 13.7%

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

   5. Applied egg-rr 14.8%

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

      1. associate-/l* 15.3%

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

      2. associate-*r/ 15.4%

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

   7. Simplified 15.4%

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

   8. Taylor expanded in k around inf 36.8%

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

      1. associate-/l* 37.3%

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

   10. Simplified 37.3%

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

       1. associate-*l/ 37.4%

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

   12. Applied egg-rr 37.4%

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

   if 3.6999999999999997e-17 < k

   1. Initial program 34.2%

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

      1. associate-/r* 34.2%

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

      2. *-commutative 34.2%

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

      3. associate-*l* 34.2%

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

      4. associate-*l/ 34.1%

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

      5. +-commutative 34.1%

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

      6. unpow2 34.1%

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

      7. sqr-neg 34.1%

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

      8. distribute-frac-neg 34.1%

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

      9. distribute-frac-neg 34.1%

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

      10. unpow2 34.1%

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

      11. associate--l+ 41.9%

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

      12. metadata-eval 41.9%

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

      13. +-rgt-identity 41.9%

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

      14. unpow2 41.9%

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

      15. distribute-frac-neg 41.9%

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

      16. distribute-frac-neg 41.9%

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

   3. Simplified 41.9%

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

   4. Taylor expanded in k around inf 73.5%

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

      1. expm1-log1p-u 62.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} - 1\right)} \]

      3. times-frac 55.6%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 59.7%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-log1p 71.3%

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

      3. times-frac 73.5%

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

      4. associate-*r* 73.5%

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

      5. times-frac 73.5%

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

      6. *-commutative 73.5%

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

   8. Simplified 73.5%

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

      1. expm1-log1p-u 64.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      2. expm1-udef 57.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)} - 1\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      3. pow2 57.9%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \ell}}{t \cdot {k}^{2}}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      4. add-sqr-sqrt 56.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{\ell \cdot \ell}{t \cdot {k}^{2}}} \cdot \sqrt{\frac{\ell \cdot \ell}{t \cdot {k}^{2}}}}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      5. pow2 56.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{\ell \cdot \ell}{t \cdot {k}^{2}}}\right)}^{2}}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      6. sqrt-div 29.2%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{\sqrt{\ell \cdot \ell}}{\sqrt{t \cdot {k}^{2}}}\right)}}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      7. sqrt-prod 12.5%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{t \cdot {k}^{2}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      8. add-sqr-sqrt 33.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{\ell}}{\sqrt{t \cdot {k}^{2}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      9. *-commutative 33.7%

         \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\sqrt{\color{blue}{{k}^{2} \cdot t}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      10. sqrt-prod 33.7%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{t}}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      11. unpow2 33.7%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      12. sqrt-prod 37.0%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

      13. add-sqr-sqrt 37.0%

          \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{k} \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   10. Applied egg-rr 37.0%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}}\right)}^{2}\right)} - 1\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]
   11. Step-by-step derivation

       1. expm1-def 46.3%

          \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}}\right)}^{2}\right)\right)} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

       2. expm1-log1p 47.3%

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

   12. Simplified 47.3%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{\frac{t}{\cos k}}}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}}\right)}^{2}\right)\\ \end{array} \]

Alternative 3: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t_m}{\cos k_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 14:\\ \;\;\;\;2 \cdot {\left(\sin k_m \cdot \left(\sqrt{t_2} \cdot \frac{k_m}{\ell}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\sin k_m \cdot \frac{k_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ t_m (cos k_m))))
   (*
    t_s
    (if (<= k_m 14.0)
      (* 2.0 (pow (* (sin k_m) (* (sqrt t_2) (/ k_m l))) -2.0))
      (/ 2.0 (* t_2 (pow (* (sin k_m) (/ k_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 t_2 = t_m / cos(k_m);
	double tmp;
	if (k_m <= 14.0) {
		tmp = 2.0 * pow((sin(k_m) * (sqrt(t_2) * (k_m / l))), -2.0);
	} else {
		tmp = 2.0 / (t_2 * pow((sin(k_m) * (k_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) :: t_2
    real(8) :: tmp
    t_2 = t_m / cos(k_m)
    if (k_m <= 14.0d0) then
        tmp = 2.0d0 * ((sin(k_m) * (sqrt(t_2) * (k_m / l))) ** (-2.0d0))
    else
        tmp = 2.0d0 / (t_2 * ((sin(k_m) * (k_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 t_2 = t_m / Math.cos(k_m);
	double tmp;
	if (k_m <= 14.0) {
		tmp = 2.0 * Math.pow((Math.sin(k_m) * (Math.sqrt(t_2) * (k_m / l))), -2.0);
	} else {
		tmp = 2.0 / (t_2 * Math.pow((Math.sin(k_m) * (k_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):
	t_2 = t_m / math.cos(k_m)
	tmp = 0
	if k_m <= 14.0:
		tmp = 2.0 * math.pow((math.sin(k_m) * (math.sqrt(t_2) * (k_m / l))), -2.0)
	else:
		tmp = 2.0 / (t_2 * math.pow((math.sin(k_m) * (k_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)
	t_2 = Float64(t_m / cos(k_m))
	tmp = 0.0
	if (k_m <= 14.0)
		tmp = Float64(2.0 * (Float64(sin(k_m) * Float64(sqrt(t_2) * Float64(k_m / l))) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(sin(k_m) * Float64(k_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)
	t_2 = t_m / cos(k_m);
	tmp = 0.0;
	if (k_m <= 14.0)
		tmp = 2.0 * ((sin(k_m) * (sqrt(t_2) * (k_m / l))) ^ -2.0);
	else
		tmp = 2.0 / (t_2 * ((sin(k_m) * (k_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_] := Block[{t$95$2 = N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 14.0], N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 14

   1. Initial program 40.1%

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

      1. associate-*l* 40.1%

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

      2. associate-*l/ 40.6%

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

      3. associate--l+ 40.6%

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

   3. Simplified 40.6%

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

      1. add-sqr-sqrt 13.6%

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

      2. pow2 13.6%

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

   5. Applied egg-rr 14.7%

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

      1. associate-/l* 15.2%

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

      2. associate-*r/ 15.2%

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

   7. Simplified 15.2%

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

   8. Taylor expanded in k around inf 36.4%

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

      1. associate-/l* 36.9%

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

   10. Simplified 36.9%

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

       1. div-inv 36.9%

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

       2. pow-flip 36.9%

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

       3. associate-/r/ 36.9%

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

       4. metadata-eval 36.9%

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

   12. Applied egg-rr 36.9%

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

       1. *-commutative 36.9%

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

       2. associate-*l* 36.9%

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

   14. Simplified 36.9%

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

   if 14 < k

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l/ 32.5%

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

      3. associate--l+ 32.5%

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

   3. Simplified 32.5%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

   5. Applied egg-rr 8.7%

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

      1. associate-/l* 8.7%

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

      2. associate-*r/ 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in k around inf 45.9%

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

      1. associate-/l* 46.0%

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

   10. Simplified 46.0%

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

       1. unpow-prod-down 42.4%

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

       2. associate-/r/ 42.3%

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

       3. pow2 42.3%

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

       4. add-sqr-sqrt 87.0%

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

   12. Applied egg-rr 87.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14:\\ \;\;\;\;2 \cdot {\left(\sin k \cdot \left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k}{\ell}\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t_m}{\cos k_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.65:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t_2} \cdot \frac{k_m}{\frac{\ell}{\sin k_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\sin k_m \cdot \frac{k_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ t_m (cos k_m))))
   (*
    t_s
    (if (<= k_m 0.65)
      (/ 2.0 (pow (* (sqrt t_2) (/ k_m (/ l (sin k_m)))) 2.0))
      (/ 2.0 (* t_2 (pow (* (sin k_m) (/ k_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 t_2 = t_m / cos(k_m);
	double tmp;
	if (k_m <= 0.65) {
		tmp = 2.0 / pow((sqrt(t_2) * (k_m / (l / sin(k_m)))), 2.0);
	} else {
		tmp = 2.0 / (t_2 * pow((sin(k_m) * (k_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) :: t_2
    real(8) :: tmp
    t_2 = t_m / cos(k_m)
    if (k_m <= 0.65d0) then
        tmp = 2.0d0 / ((sqrt(t_2) * (k_m / (l / sin(k_m)))) ** 2.0d0)
    else
        tmp = 2.0d0 / (t_2 * ((sin(k_m) * (k_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 t_2 = t_m / Math.cos(k_m);
	double tmp;
	if (k_m <= 0.65) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_2) * (k_m / (l / Math.sin(k_m)))), 2.0);
	} else {
		tmp = 2.0 / (t_2 * Math.pow((Math.sin(k_m) * (k_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):
	t_2 = t_m / math.cos(k_m)
	tmp = 0
	if k_m <= 0.65:
		tmp = 2.0 / math.pow((math.sqrt(t_2) * (k_m / (l / math.sin(k_m)))), 2.0)
	else:
		tmp = 2.0 / (t_2 * math.pow((math.sin(k_m) * (k_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)
	t_2 = Float64(t_m / cos(k_m))
	tmp = 0.0
	if (k_m <= 0.65)
		tmp = Float64(2.0 / (Float64(sqrt(t_2) * Float64(k_m / Float64(l / sin(k_m)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(sin(k_m) * Float64(k_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)
	t_2 = t_m / cos(k_m);
	tmp = 0.0;
	if (k_m <= 0.65)
		tmp = 2.0 / ((sqrt(t_2) * (k_m / (l / sin(k_m)))) ^ 2.0);
	else
		tmp = 2.0 / (t_2 * ((sin(k_m) * (k_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_] := Block[{t$95$2 = N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.65], N[(2.0 / N[Power[N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(k$95$m / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.650000000000000022

   1. Initial program 40.1%

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

      1. associate-*l* 40.1%

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

      2. associate-*l/ 40.6%

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

      3. associate--l+ 40.6%

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

   3. Simplified 40.6%

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

      1. add-sqr-sqrt 13.6%

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

      2. pow2 13.6%

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

   5. Applied egg-rr 14.7%

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

      1. associate-/l* 15.2%

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

      2. associate-*r/ 15.2%

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

   7. Simplified 15.2%

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

   8. Taylor expanded in k around inf 36.4%

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

      1. associate-/l* 36.9%

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

   10. Simplified 36.9%

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

   if 0.650000000000000022 < k

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l/ 32.5%

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

      3. associate--l+ 32.5%

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

   3. Simplified 32.5%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

   5. Applied egg-rr 8.7%

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

      1. associate-/l* 8.7%

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

      2. associate-*r/ 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in k around inf 45.9%

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

      1. associate-/l* 46.0%

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

   10. Simplified 46.0%

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

       1. unpow-prod-down 42.4%

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

       2. associate-/r/ 42.3%

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

       3. pow2 42.3%

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

       4. add-sqr-sqrt 87.0%

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

   12. Applied egg-rr 87.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.65:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t_m}{\cos k_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 14:\\ \;\;\;\;\frac{2}{{\left(\frac{k_m \cdot \sqrt{t_2}}{\frac{\ell}{\sin k_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\sin k_m \cdot \frac{k_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ t_m (cos k_m))))
   (*
    t_s
    (if (<= k_m 14.0)
      (/ 2.0 (pow (/ (* k_m (sqrt t_2)) (/ l (sin k_m))) 2.0))
      (/ 2.0 (* t_2 (pow (* (sin k_m) (/ k_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 t_2 = t_m / cos(k_m);
	double tmp;
	if (k_m <= 14.0) {
		tmp = 2.0 / pow(((k_m * sqrt(t_2)) / (l / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 / (t_2 * pow((sin(k_m) * (k_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) :: t_2
    real(8) :: tmp
    t_2 = t_m / cos(k_m)
    if (k_m <= 14.0d0) then
        tmp = 2.0d0 / (((k_m * sqrt(t_2)) / (l / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / (t_2 * ((sin(k_m) * (k_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 t_2 = t_m / Math.cos(k_m);
	double tmp;
	if (k_m <= 14.0) {
		tmp = 2.0 / Math.pow(((k_m * Math.sqrt(t_2)) / (l / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 / (t_2 * Math.pow((Math.sin(k_m) * (k_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):
	t_2 = t_m / math.cos(k_m)
	tmp = 0
	if k_m <= 14.0:
		tmp = 2.0 / math.pow(((k_m * math.sqrt(t_2)) / (l / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 / (t_2 * math.pow((math.sin(k_m) * (k_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)
	t_2 = Float64(t_m / cos(k_m))
	tmp = 0.0
	if (k_m <= 14.0)
		tmp = Float64(2.0 / (Float64(Float64(k_m * sqrt(t_2)) / Float64(l / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(sin(k_m) * Float64(k_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)
	t_2 = t_m / cos(k_m);
	tmp = 0.0;
	if (k_m <= 14.0)
		tmp = 2.0 / (((k_m * sqrt(t_2)) / (l / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 / (t_2 * ((sin(k_m) * (k_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_] := Block[{t$95$2 = N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 14.0], N[(2.0 / N[Power[N[(N[(k$95$m * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 14

   1. Initial program 40.1%

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

      1. associate-*l* 40.1%

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

      2. associate-*l/ 40.6%

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

      3. associate--l+ 40.6%

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

   3. Simplified 40.6%

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

      1. add-sqr-sqrt 13.6%

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

      2. pow2 13.6%

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

   5. Applied egg-rr 14.7%

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

      1. associate-/l* 15.2%

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

      2. associate-*r/ 15.2%

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

   7. Simplified 15.2%

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

   8. Taylor expanded in k around inf 36.4%

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

      1. associate-/l* 36.9%

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

   10. Simplified 36.9%

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

       1. associate-*l/ 36.9%

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

   12. Applied egg-rr 36.9%

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

   if 14 < k

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l/ 32.5%

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

      3. associate--l+ 32.5%

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

   3. Simplified 32.5%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

   5. Applied egg-rr 8.7%

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

      1. associate-/l* 8.7%

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

      2. associate-*r/ 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in k around inf 45.9%

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

      1. associate-/l* 46.0%

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

   10. Simplified 46.0%

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

       1. unpow-prod-down 42.4%

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

       2. associate-/r/ 42.3%

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

       3. pow2 42.3%

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

       4. add-sqr-sqrt 87.0%

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

   12. Applied egg-rr 87.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{\frac{t}{\cos k}}}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 6: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.055:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k_m \cdot \sqrt{t_m}} \cdot \left(k_m \cdot -0.08333333333333333 + \left(-0.03263888888888889 \cdot {k_m}^{3} + \frac{1}{k_m}\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m}{\cos k_m} \cdot {\left(\sin k_m \cdot \frac{k_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.055)
    (*
     2.0
     (pow
      (*
       (/ l (* k_m (sqrt t_m)))
       (+
        (* k_m -0.08333333333333333)
        (+ (* -0.03263888888888889 (pow k_m 3.0)) (/ 1.0 k_m))))
      2.0))
    (/ 2.0 (* (/ t_m (cos k_m)) (pow (* (sin k_m) (/ k_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 <= 0.055) {
		tmp = 2.0 * pow(((l / (k_m * sqrt(t_m))) * ((k_m * -0.08333333333333333) + ((-0.03263888888888889 * pow(k_m, 3.0)) + (1.0 / k_m)))), 2.0);
	} else {
		tmp = 2.0 / ((t_m / cos(k_m)) * pow((sin(k_m) * (k_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 <= 0.055d0) then
        tmp = 2.0d0 * (((l / (k_m * sqrt(t_m))) * ((k_m * (-0.08333333333333333d0)) + (((-0.03263888888888889d0) * (k_m ** 3.0d0)) + (1.0d0 / k_m)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((t_m / cos(k_m)) * ((sin(k_m) * (k_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 <= 0.055) {
		tmp = 2.0 * Math.pow(((l / (k_m * Math.sqrt(t_m))) * ((k_m * -0.08333333333333333) + ((-0.03263888888888889 * Math.pow(k_m, 3.0)) + (1.0 / k_m)))), 2.0);
	} else {
		tmp = 2.0 / ((t_m / Math.cos(k_m)) * Math.pow((Math.sin(k_m) * (k_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 <= 0.055:
		tmp = 2.0 * math.pow(((l / (k_m * math.sqrt(t_m))) * ((k_m * -0.08333333333333333) + ((-0.03263888888888889 * math.pow(k_m, 3.0)) + (1.0 / k_m)))), 2.0)
	else:
		tmp = 2.0 / ((t_m / math.cos(k_m)) * math.pow((math.sin(k_m) * (k_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 <= 0.055)
		tmp = Float64(2.0 * (Float64(Float64(l / Float64(k_m * sqrt(t_m))) * Float64(Float64(k_m * -0.08333333333333333) + Float64(Float64(-0.03263888888888889 * (k_m ^ 3.0)) + Float64(1.0 / k_m)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k_m)) * (Float64(sin(k_m) * Float64(k_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 <= 0.055)
		tmp = 2.0 * (((l / (k_m * sqrt(t_m))) * ((k_m * -0.08333333333333333) + ((-0.03263888888888889 * (k_m ^ 3.0)) + (1.0 / k_m)))) ^ 2.0);
	else
		tmp = 2.0 / ((t_m / cos(k_m)) * ((sin(k_m) * (k_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, 0.055], N[(2.0 * N[Power[N[(N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * -0.08333333333333333), $MachinePrecision] + N[(N[(-0.03263888888888889 * N[Power[k$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0550000000000000003

   1. Initial program 40.1%

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

      1. associate-/r* 40.0%

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

      2. *-commutative 40.0%

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

      3. associate-*l* 40.0%

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

      4. associate-*l/ 40.6%

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

      5. +-commutative 40.6%

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

      6. unpow2 40.6%

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

      7. sqr-neg 40.6%

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

      8. distribute-frac-neg 40.6%

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

      9. distribute-frac-neg 40.6%

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

      10. unpow2 40.6%

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

      11. associate--l+ 46.3%

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

      12. metadata-eval 46.3%

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

      13. +-rgt-identity 46.3%

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

      14. unpow2 46.3%

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

      15. distribute-frac-neg 46.3%

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

      16. distribute-frac-neg 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in k around inf 77.2%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)} \]

      2. expm1-udef 40.8%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} - 1\right)} \]

      3. times-frac 41.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 41.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 44.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. times-frac 77.2%

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

      4. associate-*r* 77.2%

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

      5. times-frac 78.7%

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

      6. *-commutative 78.7%

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

   8. Simplified 78.7%

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

      1. expm1-log1p-u 45.5%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 41.3%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} - 1\right)} \]

   10. Applied egg-rr 25.4%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 26.4%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 26.9%

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

   12. Simplified 26.9%

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

   13. Taylor expanded in k around 0 19.8%

       \[\leadsto 2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \color{blue}{\left(-0.08333333333333333 \cdot k + \left(-0.03263888888888889 \cdot {k}^{3} + \frac{1}{k}\right)\right)}\right)}^{2} \]

   if 0.0550000000000000003 < k

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l/ 32.5%

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

      3. associate--l+ 32.5%

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

   3. Simplified 32.5%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

   5. Applied egg-rr 8.7%

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

      1. associate-/l* 8.7%

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

      2. associate-*r/ 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in k around inf 45.9%

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

      1. associate-/l* 46.0%

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

   10. Simplified 46.0%

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

       1. unpow-prod-down 42.4%

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

       2. associate-/r/ 42.3%

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

       3. pow2 42.3%

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

       4. add-sqr-sqrt 87.0%

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

   12. Applied egg-rr 87.0%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.055:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \left(k \cdot -0.08333333333333333 + \left(-0.03263888888888889 \cdot {k}^{3} + \frac{1}{k}\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 7: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.014:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k_m \cdot \sqrt{t_m}} \cdot \left(k_m \cdot -0.08333333333333333 + \frac{1}{k_m}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m}{\cos k_m} \cdot {\left(\sin k_m \cdot \frac{k_m}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.014)
    (*
     2.0
     (pow
      (* (/ l (* k_m (sqrt t_m))) (+ (* k_m -0.08333333333333333) (/ 1.0 k_m)))
      2.0))
    (/ 2.0 (* (/ t_m (cos k_m)) (pow (* (sin k_m) (/ k_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 <= 0.014) {
		tmp = 2.0 * pow(((l / (k_m * sqrt(t_m))) * ((k_m * -0.08333333333333333) + (1.0 / k_m))), 2.0);
	} else {
		tmp = 2.0 / ((t_m / cos(k_m)) * pow((sin(k_m) * (k_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 <= 0.014d0) then
        tmp = 2.0d0 * (((l / (k_m * sqrt(t_m))) * ((k_m * (-0.08333333333333333d0)) + (1.0d0 / k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((t_m / cos(k_m)) * ((sin(k_m) * (k_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 <= 0.014) {
		tmp = 2.0 * Math.pow(((l / (k_m * Math.sqrt(t_m))) * ((k_m * -0.08333333333333333) + (1.0 / k_m))), 2.0);
	} else {
		tmp = 2.0 / ((t_m / Math.cos(k_m)) * Math.pow((Math.sin(k_m) * (k_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 <= 0.014:
		tmp = 2.0 * math.pow(((l / (k_m * math.sqrt(t_m))) * ((k_m * -0.08333333333333333) + (1.0 / k_m))), 2.0)
	else:
		tmp = 2.0 / ((t_m / math.cos(k_m)) * math.pow((math.sin(k_m) * (k_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 <= 0.014)
		tmp = Float64(2.0 * (Float64(Float64(l / Float64(k_m * sqrt(t_m))) * Float64(Float64(k_m * -0.08333333333333333) + Float64(1.0 / k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / cos(k_m)) * (Float64(sin(k_m) * Float64(k_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 <= 0.014)
		tmp = 2.0 * (((l / (k_m * sqrt(t_m))) * ((k_m * -0.08333333333333333) + (1.0 / k_m))) ^ 2.0);
	else
		tmp = 2.0 / ((t_m / cos(k_m)) * ((sin(k_m) * (k_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, 0.014], N[(2.0 * N[Power[N[(N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * -0.08333333333333333), $MachinePrecision] + N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0140000000000000003

   1. Initial program 40.1%

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

      1. associate-/r* 40.0%

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

      2. *-commutative 40.0%

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

      3. associate-*l* 40.0%

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

      4. associate-*l/ 40.6%

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

      5. +-commutative 40.6%

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

      6. unpow2 40.6%

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

      7. sqr-neg 40.6%

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

      8. distribute-frac-neg 40.6%

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

      9. distribute-frac-neg 40.6%

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

      10. unpow2 40.6%

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

      11. associate--l+ 46.3%

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

      12. metadata-eval 46.3%

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

      13. +-rgt-identity 46.3%

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

      14. unpow2 46.3%

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

      15. distribute-frac-neg 46.3%

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

      16. distribute-frac-neg 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in k around inf 77.2%

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

      1. expm1-log1p-u 45.0%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)} \]

      2. expm1-udef 40.8%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} - 1\right)} \]

      3. times-frac 41.0%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}}\right)} - 1\right) \]

   6. Applied egg-rr 41.0%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 44.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

      2. expm1-log1p 76.8%

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

      3. times-frac 77.2%

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

      4. associate-*r* 77.2%

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

      5. times-frac 78.7%

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

      6. *-commutative 78.7%

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

   8. Simplified 78.7%

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

      1. expm1-log1p-u 45.5%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\right)} \]

      2. expm1-udef 41.3%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} - 1\right)} \]

   10. Applied egg-rr 25.4%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 26.4%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 26.9%

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

   12. Simplified 26.9%

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

   13. Taylor expanded in k around 0 28.1%

       \[\leadsto 2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \color{blue}{\left(-0.08333333333333333 \cdot k + \frac{1}{k}\right)}\right)}^{2} \]

   if 0.0140000000000000003 < k

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l/ 32.5%

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

      3. associate--l+ 32.5%

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

   3. Simplified 32.5%

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

      1. add-sqr-sqrt 18.0%

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

      2. pow2 18.0%

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

   5. Applied egg-rr 8.7%

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

      1. associate-/l* 8.7%

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

      2. associate-*r/ 8.7%

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

   7. Simplified 8.7%

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

   8. Taylor expanded in k around inf 45.9%

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

      1. associate-/l* 46.0%

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

   10. Simplified 46.0%

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

       1. unpow-prod-down 42.4%

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

       2. associate-/r/ 42.3%

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

       3. pow2 42.3%

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

       4. add-sqr-sqrt 87.0%

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

   12. Applied egg-rr 87.0%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.014:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \left(k \cdot -0.08333333333333333 + \frac{1}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\cos k} \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 8: 76.1% accurate, 1.4× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (pow (* (sin k_m) (* (sqrt t_m) (/ k_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) {
	return t_s * (2.0 * pow((sin(k_m) * (sqrt(t_m) * (k_m / 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 * ((sin(k_m) * (sqrt(t_m) * (k_m / 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.sin(k_m) * (Math.sqrt(t_m) * (k_m / 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.sin(k_m) * (math.sqrt(t_m) * (k_m / 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(sin(k_m) * Float64(sqrt(t_m) * Float64(k_m / 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 * ((sin(k_m) * (sqrt(t_m) * (k_m / 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[Sin[k$95$m], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-*l* 37.8%

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

   2. associate-*l/ 38.1%

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

   3. associate--l+ 38.1%

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

3. Simplified 38.1%

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

   1. add-sqr-sqrt 14.9%

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

   2. pow2 14.9%

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

5. Applied egg-rr 12.8%

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

   1. associate-/l* 13.2%

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

   2. associate-*r/ 13.2%

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

7. Simplified 13.2%

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

8. Taylor expanded in k around inf 39.3%

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

   1. associate-/l* 39.7%

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

10. Simplified 39.7%

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

    1. div-inv 39.7%

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

    2. pow-flip 39.7%

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

    3. associate-/r/ 39.7%

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

    4. metadata-eval 39.7%

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

12. Applied egg-rr 39.7%

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

    1. *-commutative 39.7%

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

    2. associate-*l* 39.7%

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

14. Simplified 39.7%

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

15. Taylor expanded in k around 0 30.8%

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

16. Final simplification 30.8%

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

Alternative 9: 61.5% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (* (cos k_m) (* l l)) (* t_m (pow k_m 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 * ((cos(k_m) * (l * l)) / (t_m * pow(k_m, 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 * ((cos(k_m) * (l * l)) / (t_m * (k_m ** 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 * ((Math.cos(k_m) * (l * l)) / (t_m * Math.pow(k_m, 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 * ((math.cos(k_m) * (l * l)) / (t_m * math.pow(k_m, 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(Float64(cos(k_m) * Float64(l * l)) / Float64(t_m * (k_m ^ 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 * ((cos(k_m) * (l * l)) / (t_m * (k_m ^ 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[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.7%

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

   4. associate-*l/ 38.1%

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

   5. +-commutative 38.1%

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

   6. unpow2 38.1%

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

   7. sqr-neg 38.1%

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

   8. distribute-frac-neg 38.1%

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

   9. distribute-frac-neg 38.1%

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

   10. unpow2 38.1%

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

   11. associate--l+ 44.5%

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

   12. metadata-eval 44.5%

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

   13. +-rgt-identity 44.5%

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

   14. unpow2 44.5%

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

   15. distribute-frac-neg 44.5%

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

   16. distribute-frac-neg 44.5%

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

3. Simplified 44.5%

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

4. Taylor expanded in k around inf 75.9%

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

5. Taylor expanded in k around 0 65.3%

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

   1. pow2 61.6%

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

7. Applied egg-rr 65.3%

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

8. Final simplification 65.3%

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

Alternative 10: 73.3% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (pow (* (/ l (* k_m (sqrt t_m))) (/ 1.0 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 * sqrt(t_m))) * (1.0 / 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 * sqrt(t_m))) * (1.0d0 / 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 * Math.sqrt(t_m))) * (1.0 / 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 * math.sqrt(t_m))) * (1.0 / 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(2.0 * (Float64(Float64(l / Float64(k_m * sqrt(t_m))) * Float64(1.0 / 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 * sqrt(t_m))) * (1.0 / 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[(2.0 * N[Power[N[(N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.7%

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

   4. associate-*l/ 38.1%

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

   5. +-commutative 38.1%

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

   6. unpow2 38.1%

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

   7. sqr-neg 38.1%

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

   8. distribute-frac-neg 38.1%

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

   9. distribute-frac-neg 38.1%

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

   10. unpow2 38.1%

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

   11. associate--l+ 44.5%

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

   12. metadata-eval 44.5%

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

   13. +-rgt-identity 44.5%

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

   14. unpow2 44.5%

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

   15. distribute-frac-neg 44.5%

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

   16. distribute-frac-neg 44.5%

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

3. Simplified 44.5%

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

4. Taylor expanded in k around inf 75.9%

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

   1. expm1-log1p-u 50.3%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)} \]

   2. expm1-udef 45.4%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} - 1\right)} \]

   3. times-frac 45.5%

      \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}}\right)} - 1\right) \]

6. Applied egg-rr 45.5%

   \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 49.0%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]

   2. expm1-log1p 74.9%

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

   3. times-frac 75.9%

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

   4. associate-*r* 75.9%

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

   5. times-frac 76.9%

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

   6. *-commutative 76.9%

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

8. Simplified 76.9%

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

   1. expm1-log1p-u 50.6%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\right)} \]

   2. expm1-udef 45.7%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} - 1\right)} \]

10. Applied egg-rr 23.8%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\right)} - 1\right)} \]
11. Step-by-step derivation

    1. expm1-def 25.7%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\right)\right)} \]

    2. expm1-log1p 26.1%

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

12. Simplified 26.1%

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

13. Taylor expanded in k around 0 29.7%

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

14. Final simplification 29.7%

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

Alternative 11: 60.4% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (* (pow 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 * (2.0 * ((pow(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 * (2.0d0 * (((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 * (2.0 * ((Math.pow(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 * (2.0 * ((math.pow(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(2.0 * Float64(Float64((l ^ 2.0) * (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 * (2.0 * (((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[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.7%

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

   4. associate-*l/ 38.1%

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

   5. +-commutative 38.1%

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

   6. unpow2 38.1%

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

   7. sqr-neg 38.1%

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

   8. distribute-frac-neg 38.1%

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

   9. distribute-frac-neg 38.1%

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

   10. unpow2 38.1%

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

   11. associate--l+ 44.5%

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

   12. metadata-eval 44.5%

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

   13. +-rgt-identity 44.5%

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

   14. unpow2 44.5%

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

   15. distribute-frac-neg 44.5%

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

   16. distribute-frac-neg 44.5%

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

3. Simplified 44.5%

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

4. Taylor expanded in k around 0 63.2%

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

   1. *-commutative 63.2%

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

   2. associate-/r* 61.2%

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

6. Simplified 61.2%

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

   1. div-inv 61.2%

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

   2. pow-flip 61.6%

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

   3. metadata-eval 61.6%

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

8. Applied egg-rr 61.6%

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

   1. associate-*l/ 64.0%

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

10. Applied egg-rr 64.0%

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

11. Final simplification 64.0%

    \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t} \]

Alternative 12: 59.1% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (pow k_m -4.0) (/ (* l l) 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 * (2.0 * (pow(k_m, -4.0) * ((l * l) / 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 * (2.0d0 * ((k_m ** (-4.0d0)) * ((l * l) / 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 * (2.0 * (Math.pow(k_m, -4.0) * ((l * l) / 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 * (2.0 * (math.pow(k_m, -4.0) * ((l * l) / 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(2.0 * Float64((k_m ^ -4.0) * Float64(Float64(l * l) / 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 * (2.0 * ((k_m ^ -4.0) * ((l * l) / 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[(2.0 * N[(N[Power[k$95$m, -4.0], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

   1. associate-/r* 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*l* 37.7%

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

   4. associate-*l/ 38.1%

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

   5. +-commutative 38.1%

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

   6. unpow2 38.1%

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

   7. sqr-neg 38.1%

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

   8. distribute-frac-neg 38.1%

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

   9. distribute-frac-neg 38.1%

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

   10. unpow2 38.1%

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

   11. associate--l+ 44.5%

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

   12. metadata-eval 44.5%

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

   13. +-rgt-identity 44.5%

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

   14. unpow2 44.5%

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

   15. distribute-frac-neg 44.5%

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

   16. distribute-frac-neg 44.5%

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

3. Simplified 44.5%

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

4. Taylor expanded in k around 0 63.2%

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

   1. *-commutative 63.2%

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

   2. associate-/r* 61.2%

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

6. Simplified 61.2%

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

   1. div-inv 61.2%

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

   2. pow-flip 61.6%

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

   3. metadata-eval 61.6%

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

8. Applied egg-rr 61.6%

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

   1. pow2 61.6%

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

10. Applied egg-rr 61.6%

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

11. Final simplification 61.6%

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

Reproduce

herbie shell --seed 2023333 
(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))))