Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 99.5%

Time: 25.2s

Alternatives: 13

Speedup: 2.0×

• 39.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.0% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k\_m} \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(\sin k\_m \cdot \frac{\sqrt{t\_m}}{\sqrt{\cos k\_m}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\frac{{\sin k\_m}^{2}}{\cos k\_m}} \cdot \frac{t\_2}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (/ l k_m) (sqrt 2.0))))
   (*
    t_s
    (if (<= k_m 7.2e-11)
      (/
       2.0
       (pow (* (/ k_m l) (* (sin k_m) (/ (sqrt t_m) (sqrt (cos k_m))))) 2.0))
      (* (/ t_2 (/ (pow (sin k_m) 2.0) (cos k_m))) (/ t_2 t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = (l / k_m) * sqrt(2.0);
	double tmp;
	if (k_m <= 7.2e-11) {
		tmp = 2.0 / pow(((k_m / l) * (sin(k_m) * (sqrt(t_m) / sqrt(cos(k_m))))), 2.0);
	} else {
		tmp = (t_2 / (pow(sin(k_m), 2.0) / cos(k_m))) * (t_2 / t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (l / k_m) * sqrt(2.0d0)
    if (k_m <= 7.2d-11) then
        tmp = 2.0d0 / (((k_m / l) * (sin(k_m) * (sqrt(t_m) / sqrt(cos(k_m))))) ** 2.0d0)
    else
        tmp = (t_2 / ((sin(k_m) ** 2.0d0) / cos(k_m))) * (t_2 / t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = (l / k_m) * Math.sqrt(2.0);
	double tmp;
	if (k_m <= 7.2e-11) {
		tmp = 2.0 / Math.pow(((k_m / l) * (Math.sin(k_m) * (Math.sqrt(t_m) / Math.sqrt(Math.cos(k_m))))), 2.0);
	} else {
		tmp = (t_2 / (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))) * (t_2 / t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = (l / k_m) * math.sqrt(2.0)
	tmp = 0
	if k_m <= 7.2e-11:
		tmp = 2.0 / math.pow(((k_m / l) * (math.sin(k_m) * (math.sqrt(t_m) / math.sqrt(math.cos(k_m))))), 2.0)
	else:
		tmp = (t_2 / (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))) * (t_2 / t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(Float64(l / k_m) * sqrt(2.0))
	tmp = 0.0
	if (k_m <= 7.2e-11)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(sin(k_m) * Float64(sqrt(t_m) / sqrt(cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(t_2 / Float64((sin(k_m) ^ 2.0) / cos(k_m))) * Float64(t_2 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (l / k_m) * sqrt(2.0);
	tmp = 0.0;
	if (k_m <= 7.2e-11)
		tmp = 2.0 / (((k_m / l) * (sin(k_m) * (sqrt(t_m) / sqrt(cos(k_m))))) ^ 2.0);
	else
		tmp = (t_2 / ((sin(k_m) ^ 2.0) / cos(k_m))) * (t_2 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 7.2e-11], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999969e-11

   1. Initial program 35.7%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in t around 0 73.9%

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

      1. times-frac 76.8%

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

   7. Simplified 76.8%

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

      1. add-sqr-sqrt 41.7%

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

      2. pow2 41.7%

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

   9. Applied egg-rr 44.9%

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

       1. associate-/l* 44.8%

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

   11. Simplified 44.8%

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

   if 7.19999999999999969e-11 < k

   1. Initial program 33.3%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 71.4%

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

      1. times-frac 73.1%

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

   7. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. associate-/r* 73.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{2}{\frac{{k}^{2}}{{\ell}^{2}}}}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. add-sqr-sqrt 73.8%

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

      4. pow2 73.8%

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

      5. sqrt-div 73.8%

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

      6. unpow2 73.8%

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

      7. sqrt-prod 79.8%

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

      8. add-sqr-sqrt 79.9%

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

      9. unpow2 79.9%

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

      10. sqrt-prod 40.5%

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

      11. add-sqr-sqrt 92.6%

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

      12. associate-/l* 92.6%

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

   9. Applied egg-rr 92.6%

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

       1. div-inv 92.6%

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

       2. div-inv 92.6%

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

       3. pow-flip 93.9%

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

       4. metadata-eval 93.9%

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

   11. Applied egg-rr 93.9%

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

       1. associate-*r/ 93.9%

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

       2. *-rgt-identity 93.9%

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

   13. Simplified 93.9%

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

       1. add-sqr-sqrt 93.9%

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

       2. *-commutative 93.9%

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

       3. times-frac 93.9%

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

       4. *-commutative 93.9%

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

       5. sqrt-prod 93.9%

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

       6. sqrt-pow1 68.5%

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

       7. metadata-eval 68.5%

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

       8. unpow-1 68.5%

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

       9. clear-num 68.5%

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

       10. *-commutative 68.5%

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

       11. sqrt-prod 68.5%

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

       12. sqrt-pow1 99.4%

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

       13. metadata-eval 99.4%

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

       14. unpow-1 99.4%

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

       15. clear-num 99.4%

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

   15. Applied egg-rr 99.4%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \frac{\sqrt{t}}{\sqrt{\cos k}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \sqrt{2}}{\frac{{\sin k}^{2}}{\cos k}} \cdot \frac{\frac{\ell}{k} \cdot \sqrt{2}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 14.5:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(\sin k\_m \cdot \frac{\sqrt{t\_m}}{\sqrt{\cos k\_m}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)}{t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

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 14.5)
    (/
     2.0
     (pow (* (/ k_m l) (* (sin k_m) (/ (sqrt t_m) (sqrt (cos k_m))))) 2.0))
    (/
     (* 2.0 (* (/ l k_m) (/ l k_m)))
     (* t_m (/ (pow (sin k_m) 2.0) (cos k_m)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 14.5) {
		tmp = 2.0 / pow(((k_m / l) * (sin(k_m) * (sqrt(t_m) / sqrt(cos(k_m))))), 2.0);
	} else {
		tmp = (2.0 * ((l / k_m) * (l / k_m))) / (t_m * (pow(sin(k_m), 2.0) / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 14.5d0) then
        tmp = 2.0d0 / (((k_m / l) * (sin(k_m) * (sqrt(t_m) / sqrt(cos(k_m))))) ** 2.0d0)
    else
        tmp = (2.0d0 * ((l / k_m) * (l / k_m))) / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 14.5) {
		tmp = 2.0 / Math.pow(((k_m / l) * (Math.sin(k_m) * (Math.sqrt(t_m) / Math.sqrt(Math.cos(k_m))))), 2.0);
	} else {
		tmp = (2.0 * ((l / k_m) * (l / k_m))) / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 14.5:
		tmp = 2.0 / math.pow(((k_m / l) * (math.sin(k_m) * (math.sqrt(t_m) / math.sqrt(math.cos(k_m))))), 2.0)
	else:
		tmp = (2.0 * ((l / k_m) * (l / k_m))) / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 14.5)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(sin(k_m) * Float64(sqrt(t_m) / sqrt(cos(k_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k_m) * Float64(l / k_m))) / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 14.5)
		tmp = 2.0 / (((k_m / l) * (sin(k_m) * (sqrt(t_m) / sqrt(cos(k_m))))) ^ 2.0);
	else
		tmp = (2.0 * ((l / k_m) * (l / k_m))) / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 14.5], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 14.5

   1. Initial program 35.2%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in t around 0 73.8%

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

      1. times-frac 76.7%

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

   7. Simplified 76.7%

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

      1. add-sqr-sqrt 41.1%

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

      2. pow2 41.1%

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

   9. Applied egg-rr 44.2%

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

       1. associate-/l* 44.1%

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

   11. Simplified 44.1%

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

   if 14.5 < k

   1. Initial program 34.9%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. times-frac 73.3%

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

   7. Simplified 73.3%

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

      1. *-un-lft-identity 73.3%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. associate-/r* 74.2%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{2}{\frac{{k}^{2}}{{\ell}^{2}}}}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. add-sqr-sqrt 74.1%

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

      4. pow2 74.1%

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

      5. sqrt-div 74.1%

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

      6. unpow2 74.1%

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

      7. sqrt-prod 80.4%

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

      8. add-sqr-sqrt 80.5%

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

      9. unpow2 80.5%

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

      10. sqrt-prod 40.8%

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

      11. add-sqr-sqrt 93.8%

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

      12. associate-/l* 93.8%

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

   9. Applied egg-rr 93.8%

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

       1. div-inv 93.8%

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

       2. div-inv 93.8%

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

       3. pow-flip 95.1%

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

       4. metadata-eval 95.1%

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

   11. Applied egg-rr 95.1%

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

       1. associate-*r/ 95.1%

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

       2. *-rgt-identity 95.1%

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

   13. Simplified 95.1%

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

       1. metadata-eval 95.1%

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

       2. pow-sqr 95.1%

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

       3. unpow-1 95.1%

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

       4. clear-num 95.1%

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

       5. unpow-1 95.1%

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

       6. clear-num 95.1%

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

   15. Applied egg-rr 95.1%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14.5:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \frac{\sqrt{t}}{\sqrt{\cos k}}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 14.5:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k\_m}^{2}}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{{\left(\frac{k\_m}{\ell}\right)}^{-2}} \cdot \frac{{k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

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 14.5)
    (* 2.0 (pow (/ (/ l (pow k_m 2.0)) (sqrt t_m)) 2.0))
    (/ 2.0 (* (/ t_m (pow (/ k_m l) -2.0)) (/ (pow k_m 2.0) (cos k_m)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 14.5) {
		tmp = 2.0 * pow(((l / pow(k_m, 2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / ((t_m / pow((k_m / l), -2.0)) * (pow(k_m, 2.0) / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 14.5d0) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((t_m / ((k_m / l) ** (-2.0d0))) * ((k_m ** 2.0d0) / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 14.5) {
		tmp = 2.0 * Math.pow(((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 / ((t_m / Math.pow((k_m / l), -2.0)) * (Math.pow(k_m, 2.0) / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 14.5:
		tmp = 2.0 * math.pow(((l / math.pow(k_m, 2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 / ((t_m / math.pow((k_m / l), -2.0)) * (math.pow(k_m, 2.0) / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 14.5)
		tmp = Float64(2.0 * (Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / (Float64(k_m / l) ^ -2.0)) * Float64((k_m ^ 2.0) / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 14.5)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 / ((t_m / ((k_m / l) ^ -2.0)) * ((k_m ^ 2.0) / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 14.5], N[(2.0 * N[Power[N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[Power[N[(k$95$m / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 14.5

   1. Initial program 35.2%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in k around 0 65.2%

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

      1. add-sqr-sqrt 41.5%

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

      2. sqrt-div 34.1%

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

      3. unpow2 34.1%

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

      4. sqrt-prod 12.6%

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

      5. add-sqr-sqrt 19.1%

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

      6. sqrt-prod 19.1%

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

      7. metadata-eval 19.1%

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

      8. pow-prod-up 19.1%

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

      9. sqrt-prod 19.1%

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

      10. add-sqr-sqrt 19.1%

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

      11. sqrt-div 19.1%

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

      12. unpow2 19.1%

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

      13. sqrt-prod 14.8%

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

      14. add-sqr-sqrt 38.4%

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

      15. sqrt-prod 38.5%

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

      16. metadata-eval 38.5%

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

      17. pow-prod-up 38.5%

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

      18. sqrt-prod 42.5%

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

      19. add-sqr-sqrt 42.5%

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

   7. Applied egg-rr 42.5%

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

      1. unpow2 42.5%

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

      2. associate-/r* 42.5%

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

   9. Simplified 42.5%

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

   if 14.5 < k

   1. Initial program 34.9%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. times-frac 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in k around 0 61.5%

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

      1. clear-num 61.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{{\ell}^{2}}{{k}^{2}}}} \cdot \frac{{k}^{2} \cdot t}{\cos k}} \]

      2. frac-times 61.5%

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

      3. *-un-lft-identity 61.5%

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

      4. *-commutative 61.5%

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

      5. unpow2 61.5%

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

      6. unpow2 61.5%

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

      7. times-frac 57.4%

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

      8. clear-num 57.4%

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

      9. unpow-1 57.4%

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

      10. clear-num 57.4%

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

      11. unpow-1 57.4%

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

      12. pow-sqr 57.4%

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

      13. metadata-eval 57.4%

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

   10. Applied egg-rr 57.4%

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

       1. times-frac 63.2%

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

   12. Simplified 63.2%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14.5:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{{\left(\frac{k}{\ell}\right)}^{-2}} \cdot \frac{{k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 1.3× speedup

Math

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

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
   (* (* (/ k_m l) (/ k_m l)) (/ (* t_m (pow (sin k_m) 2.0)) (cos k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / (((k_m / l) * (k_m / l)) * ((t_m * pow(sin(k_m), 2.0)) / cos(k_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 / l) * (k_m / l)) * ((t_m * (sin(k_m) ** 2.0d0)) / cos(k_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 / (((k_m / l) * (k_m / l)) * ((t_m * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_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 / (((k_m / l) * (k_m / l)) * ((t_m * math.pow(math.sin(k_m), 2.0)) / math.cos(k_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(Float64(k_m / l) * Float64(k_m / l)) * Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / cos(k_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 / l) * (k_m / l)) * ((t_m * (sin(k_m) ^ 2.0)) / cos(k_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[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Simplified 41.1%

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

5. Taylor expanded in t around 0 73.2%

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

   1. times-frac 75.8%

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

7. Simplified 75.8%

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

   1. add-sqr-sqrt 75.8%

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

   2. sqrt-div 75.8%

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

   3. unpow2 75.8%

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

   4. sqrt-prod 38.7%

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

   5. add-sqr-sqrt 52.3%

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

   6. unpow2 52.3%

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

   7. sqrt-prod 22.2%

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

   8. add-sqr-sqrt 52.8%

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

   9. sqrt-div 52.8%

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

   10. unpow2 52.8%

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

   11. sqrt-prod 25.2%

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

   12. add-sqr-sqrt 48.2%

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

   13. unpow2 48.2%

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

   14. sqrt-prod 38.4%

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

   15. add-sqr-sqrt 90.3%

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

9. Applied egg-rr 90.3%

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

10. Final simplification 90.3%

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

Alternative 5: 91.3% accurate, 1.3× speedup

Math

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

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 (* (/ l k_m) (/ l k_m)))
   (* t_m (/ (pow (sin k_m) 2.0) (cos k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((2.0 * ((l / k_m) * (l / k_m))) / (t_m * (pow(sin(k_m), 2.0) / cos(k_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 / k_m) * (l / k_m))) / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_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 * ((l / k_m) * (l / k_m))) / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_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 * ((l / k_m) * (l / k_m))) / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(2.0 * Float64(Float64(l / k_m) * Float64(l / k_m))) / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_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 / k_m) * (l / k_m))) / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(2.0 * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Simplified 41.1%

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

5. Taylor expanded in t around 0 73.2%

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

   1. times-frac 75.8%

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

7. Simplified 75.8%

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

   1. *-un-lft-identity 75.8%

      \[\leadsto \color{blue}{1 \cdot \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   2. associate-/r* 76.0%

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{2}{\frac{{k}^{2}}{{\ell}^{2}}}}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

   3. add-sqr-sqrt 76.0%

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

   4. pow2 76.0%

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

   5. sqrt-div 76.0%

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

   6. unpow2 76.0%

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

   7. sqrt-prod 40.5%

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

   8. add-sqr-sqrt 78.8%

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

   9. unpow2 78.8%

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

   10. sqrt-prod 38.4%

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

   11. add-sqr-sqrt 90.5%

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

   12. associate-/l* 90.5%

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

9. Applied egg-rr 90.5%

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

    1. div-inv 90.1%

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

    2. div-inv 90.1%

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

    3. pow-flip 90.8%

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

    4. metadata-eval 90.8%

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

11. Applied egg-rr 90.8%

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

    1. associate-*r/ 91.2%

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

    2. *-rgt-identity 91.2%

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

13. Simplified 91.2%

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

    1. metadata-eval 91.2%

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

    2. pow-sqr 91.2%

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

    3. unpow-1 91.2%

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

    4. clear-num 91.2%

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

    5. unpow-1 91.2%

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

    6. clear-num 91.2%

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

15. Applied egg-rr 91.2%

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

16. Final simplification 91.2%

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

Alternative 6: 76.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27000000:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k\_m}^{2}}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 27000000.0)
    (* 2.0 (pow (/ (/ l (pow k_m 2.0)) (sqrt t_m)) 2.0))
    (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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) {
	double tmp;
	if (k_m <= 27000000.0) {
		tmp = 2.0 * pow(((l / pow(k_m, 2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 27000000.0d0) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / l) * t_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 27000000.0) {
		tmp = 2.0 * Math.pow(((l / Math.pow(k_m, 2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 27000000.0:
		tmp = 2.0 * math.pow(((l / math.pow(k_m, 2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 27000000.0)
		tmp = Float64(2.0 * (Float64(Float64(l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 27000000.0)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 27000000.0], N[(2.0 * N[Power[N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e7

   1. Initial program 35.3%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. add-sqr-sqrt 41.6%

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

      2. sqrt-div 33.7%

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

      3. unpow2 33.7%

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

      4. sqrt-prod 12.5%

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

      5. add-sqr-sqrt 19.4%

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

      6. sqrt-prod 19.4%

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

      7. metadata-eval 19.4%

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

      8. pow-prod-up 19.4%

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

      9. sqrt-prod 19.4%

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

      10. add-sqr-sqrt 19.4%

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

      11. sqrt-div 19.4%

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

      12. unpow2 19.4%

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

      13. sqrt-prod 14.6%

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

      14. add-sqr-sqrt 38.0%

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

      15. sqrt-prod 38.1%

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

      16. metadata-eval 38.1%

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

      17. pow-prod-up 38.1%

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

      18. sqrt-prod 42.0%

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

      19. add-sqr-sqrt 42.0%

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

   7. Applied egg-rr 42.0%

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

      1. unpow2 42.0%

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

      2. associate-/r* 42.0%

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

   9. Simplified 42.0%

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

   if 2.7e7 < k

   1. Initial program 34.4%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in k around 0 50.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Taylor expanded in k around inf 57.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-lft-identity 57.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

      2. associate-*l/ 57.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

      3. associate-/r/ 57.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      4. *-commutative 57.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      5. associate-/l* 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. unpow2 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

      7. unpow2 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. times-frac 61.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

      9. unpow2 61.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 60.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

      2. pow1/3 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. *-commutative 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 58.9%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 60.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

       2. rem-cbrt-cube 61.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

       3. pow2 61.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

       4. associate-*l* 61.7%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

   12. Applied egg-rr 61.7%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 27000000:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{{k}^{2}}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\frac{k\_m \cdot \frac{k\_m}{\ell}}{\ell \cdot \frac{\cos k\_m}{t\_m \cdot {k\_m}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 5e+56)
    (/ 2.0 (/ (* k_m (/ k_m l)) (* l (/ (cos k_m) (* t_m (pow k_m 2.0))))))
    (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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) {
	double tmp;
	if (k_m <= 5e+56) {
		tmp = 2.0 / ((k_m * (k_m / l)) / (l * (cos(k_m) / (t_m * pow(k_m, 2.0)))));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d+56) then
        tmp = 2.0d0 / ((k_m * (k_m / l)) / (l * (cos(k_m) / (t_m * (k_m ** 2.0d0)))))
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / l) * t_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e+56) {
		tmp = 2.0 / ((k_m * (k_m / l)) / (l * (Math.cos(k_m) / (t_m * Math.pow(k_m, 2.0)))));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5e+56:
		tmp = 2.0 / ((k_m * (k_m / l)) / (l * (math.cos(k_m) / (t_m * math.pow(k_m, 2.0)))))
	else:
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5e+56)
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m / l)) / Float64(l * Float64(cos(k_m) / Float64(t_m * (k_m ^ 2.0))))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e+56)
		tmp = 2.0 / ((k_m * (k_m / l)) / (l * (cos(k_m) / (t_m * (k_m ^ 2.0)))));
	else
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5e+56], N[(2.0 / N[(N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000024e56

   1. Initial program 35.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in t around 0 75.0%

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

      1. times-frac 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in k around 0 72.7%

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

      1. unpow2 72.7%

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

      2. unpow2 72.7%

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

      3. frac-times 78.7%

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

      4. pow2 78.7%

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

      5. *-commutative 78.7%

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

      6. clear-num 78.3%

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

      7. pow2 78.3%

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

      8. associate-*r/ 78.3%

         \[\leadsto \frac{2}{\frac{1}{\frac{\cos k}{{k}^{2} \cdot t}} \cdot \color{blue}{\frac{\frac{k}{\ell} \cdot k}{\ell}}} \]

      9. frac-times 79.0%

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

      10. *-un-lft-identity 79.0%

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

      11. *-commutative 79.0%

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

   10. Applied egg-rr 79.0%

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

   if 5.00000000000000024e56 < k

   1. Initial program 33.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 52.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Taylor expanded in k around inf 58.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-lft-identity 58.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

      2. associate-*l/ 58.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

      3. associate-/r/ 58.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      4. *-commutative 58.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      5. associate-/l* 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. unpow2 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

      7. unpow2 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. times-frac 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

      9. unpow2 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

   8. Simplified 63.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 62.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

      2. pow1/3 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. *-commutative 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 60.8%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 62.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

       2. rem-cbrt-cube 63.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

       3. pow2 63.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

       4. associate-*l* 63.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

   12. Applied egg-rr 63.6%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{k}{\ell}}{\ell \cdot \frac{\cos k}{t \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m}{\frac{\ell}{k\_m}}}{\ell \cdot \frac{\cos k\_m}{t\_m \cdot {k\_m}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 5e+56)
    (/ 2.0 (/ (/ k_m (/ l k_m)) (* l (/ (cos k_m) (* t_m (pow k_m 2.0))))))
    (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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) {
	double tmp;
	if (k_m <= 5e+56) {
		tmp = 2.0 / ((k_m / (l / k_m)) / (l * (cos(k_m) / (t_m * pow(k_m, 2.0)))));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d+56) then
        tmp = 2.0d0 / ((k_m / (l / k_m)) / (l * (cos(k_m) / (t_m * (k_m ** 2.0d0)))))
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / l) * t_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e+56) {
		tmp = 2.0 / ((k_m / (l / k_m)) / (l * (Math.cos(k_m) / (t_m * Math.pow(k_m, 2.0)))));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5e+56:
		tmp = 2.0 / ((k_m / (l / k_m)) / (l * (math.cos(k_m) / (t_m * math.pow(k_m, 2.0)))))
	else:
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5e+56)
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(l / k_m)) / Float64(l * Float64(cos(k_m) / Float64(t_m * (k_m ^ 2.0))))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e+56)
		tmp = 2.0 / ((k_m / (l / k_m)) / (l * (cos(k_m) / (t_m * (k_m ^ 2.0)))));
	else
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5e+56], N[(2.0 / N[(N[(k$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000024e56

   1. Initial program 35.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in t around 0 75.0%

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

      1. times-frac 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in k around 0 72.7%

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

      1. unpow2 72.7%

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

      2. unpow2 72.7%

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

      3. frac-times 78.7%

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

      4. pow2 78.7%

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

      5. *-commutative 78.7%

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

      6. clear-num 78.3%

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

      7. pow2 78.3%

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

      8. associate-*l/ 78.3%

         \[\leadsto \frac{2}{\frac{1}{\frac{\cos k}{{k}^{2} \cdot t}} \cdot \color{blue}{\frac{k \cdot \frac{k}{\ell}}{\ell}}} \]

      9. frac-times 79.0%

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

      10. *-un-lft-identity 79.0%

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

      11. clear-num 79.0%

          \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\frac{1}{\frac{\ell}{k}}}}{\frac{\cos k}{{k}^{2} \cdot t} \cdot \ell}} \]

      12. clear-num 79.0%

          \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\color{blue}{\frac{1}{\frac{k}{\ell}}}}}{\frac{\cos k}{{k}^{2} \cdot t} \cdot \ell}} \]

      13. unpow-1 79.0%

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

      14. metadata-eval 79.0%

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

      15. sqrt-pow1 50.1%

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

      16. un-div-inv 50.1%

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

      17. sqrt-pow1 79.0%

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

      18. metadata-eval 79.0%

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

      19. unpow-1 79.0%

          \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\frac{1}{\frac{k}{\ell}}}}}{\frac{\cos k}{{k}^{2} \cdot t} \cdot \ell}} \]

      20. clear-num 79.0%

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

      21. *-commutative 79.0%

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

   10. Applied egg-rr 79.0%

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

   if 5.00000000000000024e56 < k

   1. Initial program 33.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 52.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Taylor expanded in k around inf 58.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-lft-identity 58.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

      2. associate-*l/ 58.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

      3. associate-/r/ 58.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      4. *-commutative 58.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      5. associate-/l* 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. unpow2 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

      7. unpow2 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. times-frac 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

      9. unpow2 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

   8. Simplified 63.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 62.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

      2. pow1/3 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. *-commutative 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 60.8%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 62.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

       2. rem-cbrt-cube 63.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

       3. pow2 63.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

       4. associate-*l* 63.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

   12. Applied egg-rr 63.6%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\ell}{k}}}{\ell \cdot \frac{\cos k}{t \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27000000:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k\_m}{\ell}\right)}^{-2}}{t\_m \cdot {k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 27000000.0)
    (/ (* 2.0 (pow (/ k_m l) -2.0)) (* t_m (pow k_m 2.0)))
    (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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) {
	double tmp;
	if (k_m <= 27000000.0) {
		tmp = (2.0 * pow((k_m / l), -2.0)) / (t_m * pow(k_m, 2.0));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 27000000.0d0) then
        tmp = (2.0d0 * ((k_m / l) ** (-2.0d0))) / (t_m * (k_m ** 2.0d0))
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / l) * t_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 27000000.0) {
		tmp = (2.0 * Math.pow((k_m / l), -2.0)) / (t_m * Math.pow(k_m, 2.0));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 27000000.0:
		tmp = (2.0 * math.pow((k_m / l), -2.0)) / (t_m * math.pow(k_m, 2.0))
	else:
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 27000000.0)
		tmp = Float64(Float64(2.0 * (Float64(k_m / l) ^ -2.0)) / Float64(t_m * (k_m ^ 2.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 27000000.0)
		tmp = (2.0 * ((k_m / l) ^ -2.0)) / (t_m * (k_m ^ 2.0));
	else
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 27000000.0], N[(N[(2.0 * N[Power[N[(k$95$m / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e7

   1. Initial program 35.3%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around 0 74.1%

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

      1. times-frac 76.9%

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

   7. Simplified 76.9%

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

      1. *-un-lft-identity 76.9%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      2. associate-/r* 76.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{2}{\frac{{k}^{2}}{{\ell}^{2}}}}{\frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

      3. add-sqr-sqrt 76.9%

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

      4. pow2 76.9%

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

      5. sqrt-div 76.8%

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

      6. unpow2 76.8%

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

      7. sqrt-prod 27.9%

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

      8. add-sqr-sqrt 78.4%

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

      9. unpow2 78.4%

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

      10. sqrt-prod 37.7%

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

      11. add-sqr-sqrt 89.5%

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

      12. associate-/l* 89.4%

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

   9. Applied egg-rr 89.4%

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

       1. div-inv 89.0%

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

       2. div-inv 89.0%

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

       3. pow-flip 89.5%

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

       4. metadata-eval 89.5%

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

   11. Applied egg-rr 89.5%

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

       1. associate-*r/ 90.0%

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

       2. *-rgt-identity 90.0%

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

   13. Simplified 90.0%

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

   14. Taylor expanded in k around 0 77.6%

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

   if 2.7e7 < k

   1. Initial program 34.4%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in k around 0 50.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Taylor expanded in k around inf 57.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-lft-identity 57.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

      2. associate-*l/ 57.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

      3. associate-/r/ 57.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      4. *-commutative 57.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      5. associate-/l* 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. unpow2 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

      7. unpow2 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. times-frac 61.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

      9. unpow2 61.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 60.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

      2. pow1/3 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. *-commutative 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 58.9%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 60.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

       2. rem-cbrt-cube 61.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

       3. pow2 61.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

       4. associate-*l* 61.7%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

   12. Applied egg-rr 61.7%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 27000000:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{\ell}\right)}^{-2}}{t \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27000000:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 27000000.0)
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0))))
    (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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) {
	double tmp;
	if (k_m <= 27000000.0) {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 27000000.0d0) then
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0)))
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / l) * t_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 27000000.0) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 27000000.0:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 27000000.0)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 27000000.0)
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0)));
	else
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 27000000.0], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e7

   1. Initial program 35.3%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in k around 0 65.1%

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

   if 2.7e7 < k

   1. Initial program 34.4%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in k around 0 50.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Taylor expanded in k around inf 57.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-lft-identity 57.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

      2. associate-*l/ 57.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

      3. associate-/r/ 57.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      4. *-commutative 57.0%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      5. associate-/l* 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. unpow2 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

      7. unpow2 57.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. times-frac 61.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

      9. unpow2 61.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 60.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

      2. pow1/3 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. *-commutative 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 58.9%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 60.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

       2. rem-cbrt-cube 61.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

       3. pow2 61.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

       4. associate-*l* 61.7%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

   12. Applied egg-rr 61.7%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 27000000:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{{\ell}^{2}}{t\_m} \cdot \frac{2}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 5e+56)
    (* (/ (pow l 2.0) t_m) (/ 2.0 (pow k_m 4.0)))
    (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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) {
	double tmp;
	if (k_m <= 5e+56) {
		tmp = (pow(l, 2.0) / t_m) * (2.0 / pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d+56) then
        tmp = ((l ** 2.0d0) / t_m) * (2.0d0 / (k_m ** 4.0d0))
    else
        tmp = (-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / l) * t_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e+56) {
		tmp = (Math.pow(l, 2.0) / t_m) * (2.0 / Math.pow(k_m, 4.0));
	} else {
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5e+56:
		tmp = (math.pow(l, 2.0) / t_m) * (2.0 / math.pow(k_m, 4.0))
	else:
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5e+56)
		tmp = Float64(Float64((l ^ 2.0) / t_m) * Float64(2.0 / (k_m ^ 4.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / l) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e+56)
		tmp = ((l ^ 2.0) / t_m) * (2.0 / (k_m ^ 4.0));
	else
		tmp = -0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / l) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5e+56], N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000024e56

   1. Initial program 35.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in k around 0 64.4%

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

      1. associate-*r/ 64.4%

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

      2. *-commutative 64.4%

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

      3. *-commutative 64.4%

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

      4. times-frac 65.3%

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

   7. Simplified 65.3%

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

   if 5.00000000000000024e56 < k

   1. Initial program 33.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 52.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Taylor expanded in k around inf 58.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   7. Step-by-step derivation

      1. *-lft-identity 58.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

      2. associate-*l/ 58.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

      3. associate-/r/ 58.4%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      4. *-commutative 58.4%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

      5. associate-/l* 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

      6. unpow2 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

      7. unpow2 58.9%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

      8. times-frac 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

      9. unpow2 63.3%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

   8. Simplified 63.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 62.6%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

      2. pow1/3 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

      3. pow3 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

      4. *-commutative 60.8%

         \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 60.8%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. unpow1/3 62.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

       2. rem-cbrt-cube 63.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

       3. pow2 63.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

       4. associate-*l* 63.6%

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

   12. Applied egg-rr 63.6%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{{\ell}^{2}}{t} \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.2% accurate, 32.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.3333333333333333 \cdot \left(\frac{\ell}{k\_m} \cdot \frac{1}{\frac{t\_m}{\frac{\ell}{k\_m}}}\right)\right) \end{array} \]

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 (* -0.3333333333333333 (* (/ l k_m) (/ 1.0 (/ t_m (/ l k_m)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (-0.3333333333333333 * ((l / k_m) * (1.0 / (t_m / (l / k_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 * ((-0.3333333333333333d0) * ((l / k_m) * (1.0d0 / (t_m / (l / k_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 * (-0.3333333333333333 * ((l / k_m) * (1.0 / (t_m / (l / k_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 * (-0.3333333333333333 * ((l / k_m) * (1.0 / (t_m / (l / k_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(-0.3333333333333333 * Float64(Float64(l / k_m) * Float64(1.0 / Float64(t_m / Float64(l / k_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 * (-0.3333333333333333 * ((l / k_m) * (1.0 / (t_m / (l / k_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[(-0.3333333333333333 * N[(N[(l / k$95$m), $MachinePrecision] * N[(1.0 / N[(t$95$m / N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Simplified 41.1%

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

5. Taylor expanded in k around 0 34.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

6. Taylor expanded in k around inf 31.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

   1. *-lft-identity 31.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

   2. associate-*l/ 31.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

   3. associate-/r/ 31.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

   4. *-commutative 31.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

   5. associate-/l* 31.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

   6. unpow2 31.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

   7. unpow2 31.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

   8. times-frac 33.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   9. unpow2 33.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

8. Simplified 33.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 33.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

10. Applied egg-rr 33.4%

    \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]
11. Step-by-step derivation

    1. inv-pow 33.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}^{-1}} \]

    2. associate-*r* 33.6%

       \[\leadsto -0.3333333333333333 \cdot {\color{blue}{\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{k}{\ell}\right)}}^{-1} \]

    3. unpow-prod-down 33.6%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(t \cdot \frac{k}{\ell}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right)} \]

    4. clear-num 33.6%

       \[\leadsto -0.3333333333333333 \cdot \left({\left(t \cdot \color{blue}{\frac{1}{\frac{\ell}{k}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    5. clear-num 33.6%

       \[\leadsto -0.3333333333333333 \cdot \left({\left(t \cdot \frac{1}{\color{blue}{\frac{1}{\frac{k}{\ell}}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    6. unpow-1 33.6%

       \[\leadsto -0.3333333333333333 \cdot \left({\left(t \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{-1}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    7. metadata-eval 33.6%

       \[\leadsto -0.3333333333333333 \cdot \left({\left(t \cdot \frac{1}{{\left(\frac{k}{\ell}\right)}^{\color{blue}{\left(\frac{-2}{2}\right)}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    8. sqrt-pow1 45.4%

       \[\leadsto -0.3333333333333333 \cdot \left({\left(t \cdot \frac{1}{\color{blue}{\sqrt{{\left(\frac{k}{\ell}\right)}^{-2}}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    9. un-div-inv 45.4%

       \[\leadsto -0.3333333333333333 \cdot \left({\color{blue}{\left(\frac{t}{\sqrt{{\left(\frac{k}{\ell}\right)}^{-2}}}\right)}}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    10. sqrt-pow1 33.6%

        \[\leadsto -0.3333333333333333 \cdot \left({\left(\frac{t}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{\left(\frac{-2}{2}\right)}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    11. metadata-eval 33.6%

        \[\leadsto -0.3333333333333333 \cdot \left({\left(\frac{t}{{\left(\frac{k}{\ell}\right)}^{\color{blue}{-1}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    12. unpow-1 33.6%

        \[\leadsto -0.3333333333333333 \cdot \left({\left(\frac{t}{\color{blue}{\frac{1}{\frac{k}{\ell}}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    13. clear-num 33.6%

        \[\leadsto -0.3333333333333333 \cdot \left({\left(\frac{t}{\color{blue}{\frac{\ell}{k}}}\right)}^{-1} \cdot {\left(\frac{k}{\ell}\right)}^{-1}\right) \]

    14. unpow-1 33.6%

        \[\leadsto -0.3333333333333333 \cdot \left({\left(\frac{t}{\frac{\ell}{k}}\right)}^{-1} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}}\right) \]

    15. clear-num 33.6%

        \[\leadsto -0.3333333333333333 \cdot \left({\left(\frac{t}{\frac{\ell}{k}}\right)}^{-1} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

12. Applied egg-rr 33.6%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\frac{t}{\frac{\ell}{k}}\right)}^{-1} \cdot \frac{\ell}{k}\right)} \]
13. Step-by-step derivation

    1. *-commutative 33.6%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot {\left(\frac{t}{\frac{\ell}{k}}\right)}^{-1}\right)} \]

    2. unpow-1 33.6%

       \[\leadsto -0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{1}{\frac{t}{\frac{\ell}{k}}}}\right) \]

14. Simplified 33.6%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{1}{\frac{t}{\frac{\ell}{k}}}\right)} \]

15. Final simplification 33.6%

    \[\leadsto -0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\frac{t}{\frac{\ell}{k}}}\right) \]
16. Add Preprocessing

Alternative 13: 31.2% accurate, 32.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(-0.3333333333333333 \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot t\_m\right)}\right) \end{array} \]

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 (* -0.3333333333333333 (/ 1.0 (* (/ k_m l) (* (/ k_m 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 * (-0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / 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 * ((-0.3333333333333333d0) * (1.0d0 / ((k_m / l) * ((k_m / 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 * (-0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / 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 * (-0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / 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(-0.3333333333333333 * Float64(1.0 / Float64(Float64(k_m / l) * Float64(Float64(k_m / 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 * (-0.3333333333333333 * (1.0 / ((k_m / l) * ((k_m / 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[(-0.3333333333333333 * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Simplified 41.1%

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

5. Taylor expanded in k around 0 34.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

6. Taylor expanded in k around inf 31.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

   1. *-lft-identity 31.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

   2. associate-*l/ 31.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} \cdot {\ell}^{2}\right)} \]

   3. associate-/r/ 31.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

   4. *-commutative 31.7%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

   5. associate-/l* 31.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

   6. unpow2 31.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{\color{blue}{k \cdot k}}{{\ell}^{2}}} \]

   7. unpow2 31.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}} \]

   8. times-frac 33.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   9. unpow2 33.4%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{t \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

8. Simplified 33.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{1}{t \cdot {\left(\frac{k}{\ell}\right)}^{2}}} \]
9. Step-by-step derivation

   1. add-cbrt-cube 32.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}}} \]

   2. pow1/3 45.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\left(\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}^{0.3333333333333333}}} \]

   3. pow3 45.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left({\left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333}} \]

   4. *-commutative 45.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{{\left({\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}}^{3}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 45.1%

    \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left({\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}\right)}^{0.3333333333333333}}} \]
11. Step-by-step derivation

    1. unpow1/3 32.8%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot t\right)}^{3}}}} \]

    2. rem-cbrt-cube 33.4%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot t}} \]

    3. pow2 33.4%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t} \]

    4. associate-*l* 33.6%

       \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

12. Applied egg-rr 33.6%

    \[\leadsto -0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)}} \]

13. Final simplification 33.6%

    \[\leadsto -0.3333333333333333 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot t\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))