Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 94.5%

Time: 17.5s

Alternatives: 8

Speedup: 3.8×

• Could not converge on a ground truth

  1. t = 1.4135893260485003e+162
  2. l = 7.178649419540109e+270
  3. k = 2375203111060964400.0

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{-194}:\\ \;\;\;\;\ell \cdot \left(\left(\frac{2}{t} \cdot {k\_m}^{-4}\right) \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k\_m}} \cdot \sqrt[3]{t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5e-194)
   (* l (* (* (/ 2.0 t) (pow k_m -4.0)) l))
   (pow
    (/
     (pow (* (cbrt (/ (sqrt 2.0) k_m)) (cbrt t)) 2.0)
     (* (* t (pow (cbrt l) -2.0)) (* (cbrt (tan k_m)) (cbrt (sin k_m)))))
    3.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-194) {
		tmp = l * (((2.0 / t) * pow(k_m, -4.0)) * l);
	} else {
		tmp = pow((pow((cbrt((sqrt(2.0) / k_m)) * cbrt(t)), 2.0) / ((t * pow(cbrt(l), -2.0)) * (cbrt(tan(k_m)) * cbrt(sin(k_m))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-194) {
		tmp = l * (((2.0 / t) * Math.pow(k_m, -4.0)) * l);
	} else {
		tmp = Math.pow((Math.pow((Math.cbrt((Math.sqrt(2.0) / k_m)) * Math.cbrt(t)), 2.0) / ((t * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-194)
		tmp = Float64(l * Float64(Float64(Float64(2.0 / t) * (k_m ^ -4.0)) * l));
	else
		tmp = Float64((Float64(cbrt(Float64(sqrt(2.0) / k_m)) * cbrt(t)) ^ 2.0) / Float64(Float64(t * (cbrt(l) ^ -2.0)) * Float64(cbrt(tan(k_m)) * cbrt(sin(k_m))))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5e-194], N[(l * N[(N[(N[(2.0 / t), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000002e-194

   1. Initial program 37.6%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in k around 0 65.9%

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

      1. add-log-exp 63.5%

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

      2. exp-prod 63.2%

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

      3. associate-/r* 63.2%

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

      4. pow2 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. pow-exp 63.5%

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

      2. add-log-exp 65.9%

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

      3. associate-/l/ 65.9%

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

      4. rem-cbrt-cube 64.0%

         \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{{\left(t \cdot {k}^{4}\right)}^{3}}}} \cdot {\ell}^{2} \]

      5. unpow1/3 38.7%

         \[\leadsto \frac{2}{\color{blue}{{\left({\left(t \cdot {k}^{4}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot {\ell}^{2} \]

      6. pow2 38.7%

         \[\leadsto \frac{2}{{\left({\left(t \cdot {k}^{4}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      7. associate-*r* 45.7%

         \[\leadsto \color{blue}{\left(\frac{2}{{\left({\left(t \cdot {k}^{4}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \ell\right) \cdot \ell} \]

      8. unpow1/3 73.0%

         \[\leadsto \left(\frac{2}{\color{blue}{\sqrt[3]{{\left(t \cdot {k}^{4}\right)}^{3}}}} \cdot \ell\right) \cdot \ell \]

      9. rem-cbrt-cube 76.1%

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

      10. *-commutative 76.1%

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

      11. associate-/l/ 76.1%

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

      12. div-inv 76.1%

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

      13. pow-flip 76.1%

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

      14. metadata-eval 76.1%

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

   9. Applied egg-rr 76.1%

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

   if 5.0000000000000002e-194 < k

   1. Initial program 35.5%

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

      1. *-commutative 35.5%

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

      2. associate-/r* 35.5%

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

   3. Simplified 44.1%

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

      1. add-sqr-sqrt 44.1%

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

      2. add-cube-cbrt 44.1%

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

      3. times-frac 44.1%

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

   6. Applied egg-rr 86.1%

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

      1. add-cube-cbrt 86.1%

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

      2. pow3 86.1%

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

   8. Applied egg-rr 90.6%

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

      1. cbrt-prod 93.8%

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

   10. Applied egg-rr 93.8%

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

       1. *-commutative 93.8%

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

       2. cbrt-prod 95.8%

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

   12. Applied egg-rr 95.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-194}:\\ \;\;\;\;\ell \cdot \left(\left(\frac{2}{t} \cdot {k}^{-4}\right) \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k}} \cdot \sqrt[3]{t}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k\_m}} \cdot \sqrt[3]{t}\right)}^{2}}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.05e-7)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k_m 2.0))) 2.0)
   (/
    (pow
     (/
      (pow (* (cbrt (/ (sqrt 2.0) k_m)) (cbrt t)) 2.0)
      (* t (pow (cbrt l) -2.0)))
     3.0)
    (* (tan k_m) (sin k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.05e-7) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = pow((pow((cbrt((sqrt(2.0) / k_m)) * cbrt(t)), 2.0) / (t * pow(cbrt(l), -2.0))), 3.0) / (tan(k_m) * sin(k_m));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.05e-7) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = Math.pow((Math.pow((Math.cbrt((Math.sqrt(2.0) / k_m)) * Math.cbrt(t)), 2.0) / (t * Math.pow(Math.cbrt(l), -2.0))), 3.0) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.05e-7)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64((Float64((Float64(cbrt(Float64(sqrt(2.0) / k_m)) * cbrt(t)) ^ 2.0) / Float64(t * (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(tan(k_m) * sin(k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.05e-7], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[Power[N[(N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e-7

   1. Initial program 38.7%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in k around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-/r* 68.3%

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

   7. Simplified 68.3%

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

      1. pow2 68.3%

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

      2. add-log-exp 65.0%

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

      3. associate-/l/ 65.0%

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

      4. *-commutative 65.0%

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

      5. associate-/l/ 65.0%

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

      6. pow-exp 64.8%

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

      7. add-sqr-sqrt 38.1%

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

      8. pow2 38.1%

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

   9. Applied egg-rr 36.9%

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

   if 1.05e-7 < k

   1. Initial program 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-/r* 30.0%

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

   3. Simplified 44.1%

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

      1. add-sqr-sqrt 44.1%

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

      2. add-cube-cbrt 44.1%

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

      3. times-frac 44.1%

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

   6. Applied egg-rr 85.9%

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

      1. add-cube-cbrt 85.9%

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

      2. pow3 85.9%

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

   8. Applied egg-rr 92.5%

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

      1. associate-/r* 92.5%

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

      2. cube-div 92.5%

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

      3. associate-*l/ 92.5%

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

      4. pow3 92.5%

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

      5. add-cube-cbrt 92.5%

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

   10. Applied egg-rr 92.5%

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

       1. *-commutative 92.5%

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

       2. associate-/l* 92.6%

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

   12. Simplified 92.6%

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

       1. cbrt-prod 96.9%

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

       2. *-commutative 96.9%

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

   14. Applied egg-rr 96.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2}}{k}} \cdot \sqrt[3]{t}\right)}^{2}}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{t \cdot \frac{\sqrt{2}}{k\_m}}\right)}^{2}}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.5e-15)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k_m 2.0))) 2.0)
   (/
    (pow
     (/ (pow (cbrt (* t (/ (sqrt 2.0) k_m))) 2.0) (* t (pow (cbrt l) -2.0)))
     3.0)
    (* (tan k_m) (sin k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.5e-15) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = pow((pow(cbrt((t * (sqrt(2.0) / k_m))), 2.0) / (t * pow(cbrt(l), -2.0))), 3.0) / (tan(k_m) * sin(k_m));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.5e-15) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt((t * (Math.sqrt(2.0) / k_m))), 2.0) / (t * Math.pow(Math.cbrt(l), -2.0))), 3.0) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.5e-15)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64((Float64((cbrt(Float64(t * Float64(sqrt(2.0) / k_m))) ^ 2.0) / Float64(t * (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(tan(k_m) * sin(k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.5e-15], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[Power[N[Power[N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4999999999999998e-15

   1. Initial program 38.7%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in k around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-/r* 67.8%

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

   7. Simplified 67.8%

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

      1. pow2 67.8%

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

      2. add-log-exp 64.5%

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

      3. associate-/l/ 64.5%

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

      4. *-commutative 64.5%

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

      5. associate-/l/ 64.5%

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

      6. pow-exp 64.3%

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

      7. add-sqr-sqrt 38.1%

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

      8. pow2 38.1%

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

   9. Applied egg-rr 37.5%

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

   if 4.4999999999999998e-15 < k

   1. Initial program 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-/r* 30.2%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 43.6%

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

      2. add-cube-cbrt 43.6%

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

      3. times-frac 43.6%

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

   6. Applied egg-rr 86.6%

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

      1. add-cube-cbrt 86.6%

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

      2. pow3 86.6%

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

   8. Applied egg-rr 92.8%

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

      1. associate-/r* 92.9%

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

      2. cube-div 92.9%

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

      3. associate-*l/ 92.9%

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

      4. pow3 92.9%

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

      5. add-cube-cbrt 92.9%

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

   10. Applied egg-rr 92.9%

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

       1. *-commutative 92.9%

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

       2. associate-/l* 92.9%

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

   12. Simplified 92.9%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{t \cdot \frac{\sqrt{2}}{k}}\right)}^{2}}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\cos k\_m \cdot {k\_m}^{-2}}{t \cdot {\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.8e-7)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k_m 2.0))) 2.0)
   (*
    (* 2.0 (/ (* (cos k_m) (pow k_m -2.0)) (* t (pow (sin k_m) 2.0))))
    (* l l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.8e-7) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) * pow(k_m, -2.0)) / (t * pow(sin(k_m), 2.0)))) * (l * l);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.8d-7) then
        tmp = (l * (sqrt((2.0d0 / t)) / (k_m ** 2.0d0))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) * (k_m ** (-2.0d0))) / (t * (sin(k_m) ** 2.0d0)))) * (l * l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.8e-7) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) * Math.pow(k_m, -2.0)) / (t * Math.pow(Math.sin(k_m), 2.0)))) * (l * l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.8e-7:
		tmp = math.pow((l * (math.sqrt((2.0 / t)) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) * math.pow(k_m, -2.0)) / (t * math.pow(math.sin(k_m), 2.0)))) * (l * l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.8e-7)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * (k_m ^ -2.0)) / Float64(t * (sin(k_m) ^ 2.0)))) * Float64(l * l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.8e-7)
		tmp = (l * (sqrt((2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = (2.0 * ((cos(k_m) * (k_m ^ -2.0)) / (t * (sin(k_m) ^ 2.0)))) * (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.8e-7], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.7999999999999995e-7

   1. Initial program 38.7%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in k around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-/r* 68.3%

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

   7. Simplified 68.3%

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

      1. pow2 68.3%

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

      2. add-log-exp 65.0%

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

      3. associate-/l/ 65.0%

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

      4. *-commutative 65.0%

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

      5. associate-/l/ 65.0%

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

      6. pow-exp 64.8%

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

      7. add-sqr-sqrt 38.1%

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

      8. pow2 38.1%

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

   9. Applied egg-rr 36.9%

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

   if 5.7999999999999995e-7 < k

   1. Initial program 30.0%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around 0 80.8%

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

      1. associate-/r* 80.8%

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

   7. Simplified 80.8%

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

      1. div-inv 80.9%

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

      2. pow-flip 81.0%

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

      3. metadata-eval 81.0%

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

   9. Applied egg-rr 81.0%

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

Alternative 5: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{2}}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.5e-7)
   (pow (* l (/ (sqrt (/ 2.0 t)) (pow k_m 2.0))) 2.0)
   (/ (* 2.0 (/ (pow l 2.0) (* t (pow k_m 2.0)))) (* (tan k_m) (sin k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e-7) {
		tmp = pow((l * (sqrt((2.0 / t)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * (pow(l, 2.0) / (t * pow(k_m, 2.0)))) / (tan(k_m) * sin(k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.5d-7) then
        tmp = (l * (sqrt((2.0d0 / t)) / (k_m ** 2.0d0))) ** 2.0d0
    else
        tmp = (2.0d0 * ((l ** 2.0d0) / (t * (k_m ** 2.0d0)))) / (tan(k_m) * sin(k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e-7) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 2.0)))) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.5e-7:
		tmp = math.pow((l * (math.sqrt((2.0 / t)) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = (2.0 * (math.pow(l, 2.0) / (t * math.pow(k_m, 2.0)))) / (math.tan(k_m) * math.sin(k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.5e-7)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k_m ^ 2.0)))) / Float64(tan(k_m) * sin(k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.5e-7)
		tmp = (l * (sqrt((2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = (2.0 * ((l ^ 2.0) / (t * (k_m ^ 2.0)))) / (tan(k_m) * sin(k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.5e-7], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5000000000000003e-7

   1. Initial program 38.7%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in k around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-/r* 68.3%

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

   7. Simplified 68.3%

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

      1. pow2 68.3%

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

      2. add-log-exp 65.0%

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

      3. associate-/l/ 65.0%

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

      4. *-commutative 65.0%

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

      5. associate-/l/ 65.0%

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

      6. pow-exp 64.8%

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

      7. add-sqr-sqrt 38.1%

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

      8. pow2 38.1%

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

   9. Applied egg-rr 36.9%

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

   if 5.5000000000000003e-7 < k

   1. Initial program 30.0%

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

      1. *-commutative 30.0%

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

      2. associate-/r* 30.0%

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

   3. Simplified 44.1%

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

      1. add-sqr-sqrt 44.1%

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

      2. add-cube-cbrt 44.1%

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

      3. times-frac 44.1%

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

   6. Applied egg-rr 85.9%

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

      1. add-cube-cbrt 85.9%

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

      2. pow3 85.9%

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

   8. Applied egg-rr 92.5%

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

      1. associate-/r* 92.5%

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

      2. cube-div 92.5%

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

      3. associate-*l/ 92.5%

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

      4. pow3 92.5%

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

      5. add-cube-cbrt 92.5%

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

   10. Applied egg-rr 92.5%

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

       1. *-commutative 92.5%

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

       2. associate-/l* 92.6%

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

   12. Simplified 92.6%

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

   13. Taylor expanded in t around 0 80.8%

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

       1. *-commutative 80.8%

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

       2. unpow2 80.8%

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

       3. rem-square-sqrt 80.9%

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

       4. associate-/l* 80.9%

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

       5. *-commutative 80.9%

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

   15. Simplified 80.9%

       \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}}}{\sin k \cdot \tan k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.1% accurate, 1.4× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (pow (* l (/ (sqrt (/ 2.0 t)) (pow k_m 2.0))) 2.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return pow((l * (sqrt((2.0 / t)) / pow(k_m, 2.0))), 2.0);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * (sqrt((2.0d0 / t)) / (k_m ** 2.0d0))) ** 2.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return Math.pow((l * (Math.sqrt((2.0 / t)) / Math.pow(k_m, 2.0))), 2.0);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return math.pow((l * (math.sqrt((2.0 / t)) / math.pow(k_m, 2.0))), 2.0)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(sqrt(Float64(2.0 / t)) / (k_m ^ 2.0))) ^ 2.0
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * (sqrt((2.0 / t)) / (k_m ^ 2.0))) ^ 2.0;
end

Wolfram

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

Derivation

1. Initial program 36.7%

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

3. Simplified 47.6%

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

5. Taylor expanded in k around 0 68.1%

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

   1. *-commutative 68.1%

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

   2. associate-/r* 68.1%

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

7. Simplified 68.1%

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

   1. pow2 68.1%

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

   2. add-log-exp 65.5%

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

   3. associate-/l/ 65.5%

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

   4. *-commutative 65.5%

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

   5. associate-/l/ 65.5%

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

   6. pow-exp 64.7%

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

   7. add-sqr-sqrt 43.9%

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

   8. pow2 43.9%

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

9. Applied egg-rr 37.4%

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

Alternative 7: 68.7% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* l (* (* (/ 2.0 t) (pow k_m -4.0)) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (((2.0 / t) * pow(k_m, -4.0)) * l);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (((2.0d0 / t) * (k_m ** (-4.0d0))) * l)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (((2.0 / t) * Math.pow(k_m, -4.0)) * l);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (((2.0 / t) * math.pow(k_m, -4.0)) * l)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(Float64(Float64(2.0 / t) * (k_m ^ -4.0)) * l))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (((2.0 / t) * (k_m ^ -4.0)) * l);
end

Wolfram

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

Derivation

1. Initial program 36.7%

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

3. Simplified 47.6%

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

5. Taylor expanded in k around 0 68.1%

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

   1. add-log-exp 65.5%

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

   2. exp-prod 64.7%

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

   3. associate-/r* 64.7%

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

   4. pow2 64.7%

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

7. Applied egg-rr 64.7%

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

   1. pow-exp 65.5%

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

   2. add-log-exp 68.1%

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

   3. associate-/l/ 68.1%

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

   4. rem-cbrt-cube 65.8%

      \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{{\left(t \cdot {k}^{4}\right)}^{3}}}} \cdot {\ell}^{2} \]

   5. unpow1/3 42.9%

      \[\leadsto \frac{2}{\color{blue}{{\left({\left(t \cdot {k}^{4}\right)}^{3}\right)}^{0.3333333333333333}}} \cdot {\ell}^{2} \]

   6. pow2 42.9%

      \[\leadsto \frac{2}{{\left({\left(t \cdot {k}^{4}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   7. associate-*r* 47.6%

      \[\leadsto \color{blue}{\left(\frac{2}{{\left({\left(t \cdot {k}^{4}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \ell\right) \cdot \ell} \]

   8. unpow1/3 72.1%

      \[\leadsto \left(\frac{2}{\color{blue}{\sqrt[3]{{\left(t \cdot {k}^{4}\right)}^{3}}}} \cdot \ell\right) \cdot \ell \]

   9. rem-cbrt-cube 75.4%

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

   10. *-commutative 75.4%

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

   11. associate-/l/ 75.4%

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

   12. div-inv 75.4%

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

   13. pow-flip 75.4%

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

   14. metadata-eval 75.4%

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

9. Applied egg-rr 75.4%

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

10. Final simplification 75.4%

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

Alternative 8: 29.2% accurate, 421.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 0 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 0.0)

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 0.0;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 0.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 0.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 0.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return 0.0
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 0.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := 0.0

Derivation

1. Initial program 36.7%

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

3. Simplified 47.6%

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

5. Taylor expanded in k around 0 68.1%

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

   1. add-log-exp 65.5%

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

   2. exp-prod 64.7%

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

   3. associate-/r* 64.7%

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

   4. pow2 64.7%

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

7. Applied egg-rr 64.7%

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

8. Taylor expanded in k around inf 29.3%

   \[\leadsto \log \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 29.3%

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 29.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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))))