Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 88.9%

Time: 17.7s

Alternatives: 17

Speedup: 3.8×

• 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: 35.6% 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: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k}\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-263}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+230}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_m \cdot {\left(t\_m \cdot t\_3\right)}^{-2}\right)\right) \cdot \frac{t\_2}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k))
        (t_3 (* (pow (cbrt l) -2.0) (cbrt (* (sin k) (tan k))))))
   (*
    t_s
    (if (<= (* l l) 4e-263)
      (pow (cbrt (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))) 3.0)
      (if (<= (* l l) 1e+230)
        (*
         -2.0
         (/
          (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (- (pow l 2.0)))
          (pow k 2.0)))
        (* (* t_2 (* t_m (pow (* t_m t_3) -2.0))) (/ t_2 t_3)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sqrt(2.0) / k;
	double t_3 = pow(cbrt(l), -2.0) * cbrt((sin(k) * tan(k)));
	double tmp;
	if ((l * l) <= 4e-263) {
		tmp = pow(cbrt((2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0))), 3.0);
	} else if ((l * l) <= 1e+230) {
		tmp = -2.0 * ((((cos(k) / t_m) / pow(sin(k), 2.0)) * -pow(l, 2.0)) / pow(k, 2.0));
	} else {
		tmp = (t_2 * (t_m * pow((t_m * t_3), -2.0))) * (t_2 / t_3);
	}
	return t_s * tmp;
}

Java

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) {
	double t_2 = Math.sqrt(2.0) / k;
	double t_3 = Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if ((l * l) <= 4e-263) {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0))), 3.0);
	} else if ((l * l) <= 1e+230) {
		tmp = -2.0 * ((((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * -Math.pow(l, 2.0)) / Math.pow(k, 2.0));
	} else {
		tmp = (t_2 * (t_m * Math.pow((t_m * t_3), -2.0))) * (t_2 / t_3);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sqrt(2.0) / k)
	t_3 = Float64((cbrt(l) ^ -2.0) * cbrt(Float64(sin(k) * tan(k))))
	tmp = 0.0
	if (Float64(l * l) <= 4e-263)
		tmp = cbrt(Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0))) ^ 3.0;
	elseif (Float64(l * l) <= 1e+230)
		tmp = Float64(-2.0 * Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(-(l ^ 2.0))) / (k ^ 2.0)));
	else
		tmp = Float64(Float64(t_2 * Float64(t_m * (Float64(t_m * t_3) ^ -2.0))) * Float64(t_2 / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 4e-263], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+230], N[(-2.0 * N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-N[Power[l, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$m * N[Power[N[(t$95$m * t$95$3), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4e-263

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   6. Applied egg-rr 26.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in k around 0 44.7%

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

   if 4e-263 < (*.f64 l l) < 1.0000000000000001e230

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-/r* 36.6%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 21.5%

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

      2. pow2 21.5%

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

      3. sqrt-div 21.5%

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

      4. sqrt-pow1 23.6%

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

      5. metadata-eval 23.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 13.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 23.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 23.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.4%

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

   9. Simplified 92.4%

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

       1. pow2 92.4%

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

       2. associate-*l/ 93.5%

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

       3. pow2 93.5%

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

       4. associate-/r* 93.5%

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

       5. mul-1-neg 93.5%

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

   11. Applied egg-rr 93.5%

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

   if 1.0000000000000001e230 < (*.f64 l l)

   1. Initial program 40.7%

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

   3. Simplified 40.7%

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

   6. Applied egg-rr 82.0%

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

      1. associate-/r/ 82.0%

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

      2. associate-/r* 82.0%

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

      3. associate-/r/ 84.7%

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

   8. Simplified 84.7%

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

      1. *-commutative 84.7%

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

      2. cbrt-prod 84.6%

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

   10. Applied egg-rr 84.6%

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

       1. associate-*r/ 84.7%

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

   12. Applied egg-rr 84.7%

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

       1. associate-/l* 84.6%

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

       2. associate-*l* 87.8%

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

       3. associate-*l* 87.8%

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

       4. *-commutative 87.8%

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

       5. associate-/l* 87.8%

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

   14. Simplified 87.8%

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

       1. associate-*l/ 87.9%

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

       2. associate-/r* 87.9%

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

       3. pow1 87.9%

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

       4. pow1 87.9%

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

       5. pow-div 87.9%

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

       6. metadata-eval 87.9%

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

       7. metadata-eval 87.9%

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

   16. Applied egg-rr 87.9%

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

       1. associate-*l/ 87.8%

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

       2. associate-*r/ 87.9%

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

       3. *-rgt-identity 87.9%

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

   18. Simplified 87.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-263}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+230}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-263}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+230}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 4e-263)
    (pow (cbrt (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))) 3.0)
    (if (<= (* l l) 1e+230)
      (*
       -2.0
       (/
        (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (- (pow l 2.0)))
        (pow k 2.0)))
      (pow
       (cbrt (/ 2.0 (pow (* (* k (/ (sin k) l)) (sqrt (/ t_m (cos k)))) 2.0)))
       3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 4e-263) {
		tmp = pow(cbrt((2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0))), 3.0);
	} else if ((l * l) <= 1e+230) {
		tmp = -2.0 * ((((cos(k) / t_m) / pow(sin(k), 2.0)) * -pow(l, 2.0)) / pow(k, 2.0));
	} else {
		tmp = pow(cbrt((2.0 / pow(((k * (sin(k) / l)) * sqrt((t_m / cos(k)))), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if ((l * l) <= 4e-263) {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0))), 3.0);
	} else if ((l * l) <= 1e+230) {
		tmp = -2.0 * ((((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * -Math.pow(l, 2.0)) / Math.pow(k, 2.0));
	} else {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((k * (Math.sin(k) / l)) * Math.sqrt((t_m / Math.cos(k)))), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 4e-263)
		tmp = cbrt(Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0))) ^ 3.0;
	elseif (Float64(l * l) <= 1e+230)
		tmp = Float64(-2.0 * Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(-(l ^ 2.0))) / (k ^ 2.0)));
	else
		tmp = cbrt(Float64(2.0 / (Float64(Float64(k * Float64(sin(k) / l)) * sqrt(Float64(t_m / cos(k)))) ^ 2.0))) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 4e-263], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+230], N[(-2.0 * N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-N[Power[l, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 4e-263

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   6. Applied egg-rr 26.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in k around 0 44.7%

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

   if 4e-263 < (*.f64 l l) < 1.0000000000000001e230

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

      2. associate-/r* 36.6%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 21.5%

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

      2. pow2 21.5%

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

      3. sqrt-div 21.5%

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

      4. sqrt-pow1 23.6%

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

      5. metadata-eval 23.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 13.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 23.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 23.6%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 92.4%

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

   9. Simplified 92.4%

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

       1. pow2 92.4%

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

       2. associate-*l/ 93.5%

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

       3. pow2 93.5%

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

       4. associate-/r* 93.5%

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

       5. mul-1-neg 93.5%

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

   11. Applied egg-rr 93.5%

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

   if 1.0000000000000001e230 < (*.f64 l l)

   1. Initial program 40.7%

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

   3. Simplified 40.7%

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

   6. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in t around 0 47.0%

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

      1. associate-/l* 47.0%

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

   9. Simplified 47.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-263}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+230}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.1 \cdot 10^{-132}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+160}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 5.1e-132)
    (pow (cbrt (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))) 3.0)
    (if (<= l 1.06e+160)
      (*
       -2.0
       (/
        (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (- (pow l 2.0)))
        (pow k 2.0)))
      (/
       2.0
       (*
        (* (* (sin k) (tan k)) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))
        (* (/ k t_m) (/ k t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5.1e-132) {
		tmp = pow(cbrt((2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0))), 3.0);
	} else if (l <= 1.06e+160) {
		tmp = -2.0 * ((((cos(k) / t_m) / pow(sin(k), 2.0)) * -pow(l, 2.0)) / pow(k, 2.0));
	} else {
		tmp = 2.0 / (((sin(k) * tan(k)) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)) * ((k / t_m) * (k / t_m)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (l <= 5.1e-132) {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0))), 3.0);
	} else if (l <= 1.06e+160) {
		tmp = -2.0 * ((((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * -Math.pow(l, 2.0)) / Math.pow(k, 2.0));
	} else {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)) * ((k / t_m) * (k / t_m)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 5.1e-132)
		tmp = cbrt(Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0))) ^ 3.0;
	elseif (l <= 1.06e+160)
		tmp = Float64(-2.0 * Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(-(l ^ 2.0))) / (k ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0)) * Float64(Float64(k / t_m) * Float64(k / t_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[l, 5.1e-132], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, 1.06e+160], N[(-2.0 * N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-N[Power[l, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 5.10000000000000005e-132

   1. Initial program 30.9%

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

   3. Simplified 30.9%

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

   6. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in k around 0 40.9%

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

   if 5.10000000000000005e-132 < l < 1.0599999999999999e160

   1. Initial program 32.3%

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

      1. *-commutative 32.3%

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

      2. associate-/r* 32.3%

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

   3. Simplified 43.7%

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

      1. add-sqr-sqrt 18.4%

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

      2. pow2 18.4%

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

      3. sqrt-div 18.4%

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

      4. sqrt-pow1 21.1%

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

      5. metadata-eval 21.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 21.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 21.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 21.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.9%

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

   9. Simplified 90.9%

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

       1. pow2 90.9%

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

       2. associate-*l/ 89.8%

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

       3. pow2 89.8%

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

       4. associate-/r* 89.8%

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

       5. mul-1-neg 89.8%

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

   11. Applied egg-rr 89.8%

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

   if 1.0599999999999999e160 < l

   1. Initial program 52.2%

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

   3. Simplified 52.2%

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

      1. +-commutative 52.2%

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

      2. associate-+l- 52.2%

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

      3. metadata-eval 52.2%

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

      4. --rgt-identity 52.2%

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

      5. unpow2 52.2%

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

   6. Applied egg-rr 52.2%

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

      1. add-cube-cbrt 52.2%

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

      2. associate-*l* 52.2%

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

      3. cbrt-div 52.2%

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

      4. rem-cbrt-cube 52.2%

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

      5. cbrt-prod 52.2%

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

      6. pow2 52.2%

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

      7. pow2 52.2%

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

      8. cbrt-div 52.2%

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

      9. rem-cbrt-cube 69.0%

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

      10. cbrt-prod 91.7%

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

      11. pow2 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. unpow2 91.7%

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

      2. cube-mult 91.7%

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

   10. Simplified 91.7%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.1 \cdot 10^{-132}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+160}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.4e-6)
    (pow (cbrt (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))) 3.0)
    (*
     -2.0
     (/
      (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (- (pow l 2.0)))
      (pow k 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.4e-6) {
		tmp = pow(cbrt((2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0))), 3.0);
	} else {
		tmp = -2.0 * ((((cos(k) / t_m) / pow(sin(k), 2.0)) * -pow(l, 2.0)) / pow(k, 2.0));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 1.4e-6) {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0))), 3.0);
	} else {
		tmp = -2.0 * ((((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * -Math.pow(l, 2.0)) / Math.pow(k, 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.4e-6)
		tmp = cbrt(Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(-(l ^ 2.0))) / (k ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 1.4e-6], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(-2.0 * N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-N[Power[l, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.39999999999999994e-6

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   6. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in k around 0 44.3%

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

   if 1.39999999999999994e-6 < k

   1. Initial program 25.5%

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

      1. *-commutative 25.5%

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

      2. associate-/r* 25.4%

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

   3. Simplified 40.6%

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

      1. add-sqr-sqrt 16.4%

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

      2. pow2 16.4%

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

      3. sqrt-div 16.4%

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

      4. sqrt-pow1 19.5%

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

      5. metadata-eval 19.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 13.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 23.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 23.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 72.4%

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

   9. Simplified 72.4%

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

       1. pow2 72.4%

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

       2. associate-*l/ 73.4%

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

       3. pow2 73.4%

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

       4. associate-/r* 73.4%

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

       5. mul-1-neg 73.4%

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

   11. Applied egg-rr 73.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(-{\ell}^{2}\right)}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{t\_m \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.5e-5)
    (pow (cbrt (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))) 3.0)
    (/
     2.0
     (* (pow k 2.0) (/ (* t_m (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e-5) {
		tmp = pow(cbrt((2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0))), 3.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) * ((t_m * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k))));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 2.5e-5) {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0))), 3.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t_m * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.5e-5)
		tmp = cbrt(Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2.5e-5], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000012e-5

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   6. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in k around 0 44.3%

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

   if 2.50000000000000012e-5 < k

   1. Initial program 25.5%

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

   3. Simplified 25.5%

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

      1. +-commutative 25.5%

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

      2. associate-+l- 40.6%

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

      3. metadata-eval 40.6%

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

      4. --rgt-identity 40.6%

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

      5. unpow2 40.6%

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

   6. Applied egg-rr 40.6%

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

   7. Taylor expanded in t around 0 70.7%

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

      1. associate-/l* 73.6%

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

   9. Simplified 73.6%

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

Alternative 6: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\_m\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.45e-6)
    (pow (cbrt (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))) 3.0)
    (*
     (* l l)
     (/ 2.0 (/ (* (pow (sin k) 2.0) (* (pow k 2.0) t_m)) (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.45e-6) {
		tmp = pow(cbrt((2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0))), 3.0);
	} else {
		tmp = (l * l) * (2.0 / ((pow(sin(k), 2.0) * (pow(k, 2.0) * t_m)) / cos(k)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 1.45e-6) {
		tmp = Math.pow(Math.cbrt((2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0))), 3.0);
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(Math.sin(k), 2.0) * (Math.pow(k, 2.0) * t_m)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.45e-6)
		tmp = cbrt(Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64((k ^ 2.0) * t_m)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 1.45e-6], N[Power[N[Power[N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4500000000000001e-6

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   6. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right)\right)}^{2}}}\right)}^{3}} \]

   7. Taylor expanded in k around 0 44.3%

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

   if 1.4500000000000001e-6 < k

   1. Initial program 25.5%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 70.8%

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

      1. associate-*r* 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t\_m} \cdot \frac{k}{t\_m}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\_m\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.9e-195)
    (/
     (/ 2.0 (* (/ k t_m) (/ k t_m)))
     (* (* (sin k) (tan k)) (pow (/ (pow t_m 1.5) l) 2.0)))
    (*
     (* l l)
     (/ 2.0 (/ (* (pow (sin k) 2.0) (* (pow k 2.0) t_m)) (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.9e-195) {
		tmp = (2.0 / ((k / t_m) * (k / t_m))) / ((sin(k) * tan(k)) * pow((pow(t_m, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 / ((pow(sin(k), 2.0) * (pow(k, 2.0) * t_m)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.9d-195) then
        tmp = (2.0d0 / ((k / t_m) * (k / t_m))) / ((sin(k) * tan(k)) * (((t_m ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (l * l) * (2.0d0 / (((sin(k) ** 2.0d0) * ((k ** 2.0d0) * t_m)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (l <= 1.9e-195) {
		tmp = (2.0 / ((k / t_m) * (k / t_m))) / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(Math.sin(k), 2.0) * (Math.pow(k, 2.0) * t_m)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.9e-195:
		tmp = (2.0 / ((k / t_m) * (k / t_m))) / ((math.sin(k) * math.tan(k)) * math.pow((math.pow(t_m, 1.5) / l), 2.0))
	else:
		tmp = (l * l) * (2.0 / ((math.pow(math.sin(k), 2.0) * (math.pow(k, 2.0) * t_m)) / math.cos(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.9e-195)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t_m) * Float64(k / t_m))) / Float64(Float64(sin(k) * tan(k)) * (Float64((t_m ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64((k ^ 2.0) * t_m)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.9e-195)
		tmp = (2.0 / ((k / t_m) * (k / t_m))) / ((sin(k) * tan(k)) * (((t_m ^ 1.5) / l) ^ 2.0));
	else
		tmp = (l * l) * (2.0 / (((sin(k) ^ 2.0) * ((k ^ 2.0) * t_m)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[l, 1.9e-195], N[(N[(2.0 / N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.90000000000000006e-195

   1. Initial program 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-/r* 31.1%

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

   3. Simplified 37.4%

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

      1. add-sqr-sqrt 14.6%

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

      2. pow2 14.6%

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

      3. sqrt-div 14.6%

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

      4. sqrt-pow1 19.9%

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

      5. metadata-eval 19.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 4.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 24.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 24.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. +-rgt-identity 24.3%

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

      2. unpow2 24.3%

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

   8. Applied egg-rr 24.3%

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

   if 1.90000000000000006e-195 < l

   1. Initial program 36.3%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. associate-*r* 79.8%

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

   7. Simplified 79.8%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-195}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\_m\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3.3e-195)
    (/
     2.0
     (*
      (* (/ k t_m) (/ k t_m))
      (* (* (sin k) (tan k)) (pow (/ (pow t_m 1.5) l) 2.0))))
    (*
     (* l l)
     (/ 2.0 (/ (* (pow (sin k) 2.0) (* (pow k 2.0) t_m)) (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.3e-195) {
		tmp = 2.0 / (((k / t_m) * (k / t_m)) * ((sin(k) * tan(k)) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = (l * l) * (2.0 / ((pow(sin(k), 2.0) * (pow(k, 2.0) * t_m)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.3d-195) then
        tmp = 2.0d0 / (((k / t_m) * (k / t_m)) * ((sin(k) * tan(k)) * (((t_m ** 1.5d0) / l) ** 2.0d0)))
    else
        tmp = (l * l) * (2.0d0 / (((sin(k) ** 2.0d0) * ((k ** 2.0d0) * t_m)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (l <= 3.3e-195) {
		tmp = 2.0 / (((k / t_m) * (k / t_m)) * ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(Math.sin(k), 2.0) * (Math.pow(k, 2.0) * t_m)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.3e-195:
		tmp = 2.0 / (((k / t_m) * (k / t_m)) * ((math.sin(k) * math.tan(k)) * math.pow((math.pow(t_m, 1.5) / l), 2.0)))
	else:
		tmp = (l * l) * (2.0 / ((math.pow(math.sin(k), 2.0) * (math.pow(k, 2.0) * t_m)) / math.cos(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.3e-195)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / t_m) * Float64(k / t_m)) * Float64(Float64(sin(k) * tan(k)) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64((k ^ 2.0) * t_m)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.3e-195)
		tmp = 2.0 / (((k / t_m) * (k / t_m)) * ((sin(k) * tan(k)) * (((t_m ^ 1.5) / l) ^ 2.0)));
	else
		tmp = (l * l) * (2.0 / (((sin(k) ^ 2.0) * ((k ^ 2.0) * t_m)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[l, 3.3e-195], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.3e-195

   1. Initial program 31.2%

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

   3. Simplified 31.2%

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

      1. +-commutative 31.2%

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

      2. associate-+l- 37.4%

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

      3. metadata-eval 37.4%

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

      4. --rgt-identity 37.4%

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

      5. unpow2 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. add-sqr-sqrt 14.6%

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

      2. pow2 14.6%

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

      3. sqrt-div 14.6%

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

      4. sqrt-pow1 19.9%

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

      5. metadata-eval 19.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 4.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 24.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 24.5%

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

   if 3.3e-195 < l

   1. Initial program 36.3%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. associate-*r* 79.8%

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

   7. Simplified 79.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.3 \cdot 10^{-195}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t\_m}\right)}^{2}}}{{k}^{2} \cdot {\left(\frac{1}{\frac{\ell}{{t\_m}^{1.5}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\_m\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.3e-195)
    (/
     (/ 2.0 (pow (/ k t_m) 2.0))
     (* (pow k 2.0) (pow (/ 1.0 (/ l (pow t_m 1.5))) 2.0)))
    (*
     (* l l)
     (/ 2.0 (/ (* (pow (sin k) 2.0) (* (pow k 2.0) t_m)) (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.3e-195) {
		tmp = (2.0 / pow((k / t_m), 2.0)) / (pow(k, 2.0) * pow((1.0 / (l / pow(t_m, 1.5))), 2.0));
	} else {
		tmp = (l * l) * (2.0 / ((pow(sin(k), 2.0) * (pow(k, 2.0) * t_m)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 2.3d-195) then
        tmp = (2.0d0 / ((k / t_m) ** 2.0d0)) / ((k ** 2.0d0) * ((1.0d0 / (l / (t_m ** 1.5d0))) ** 2.0d0))
    else
        tmp = (l * l) * (2.0d0 / (((sin(k) ** 2.0d0) * ((k ** 2.0d0) * t_m)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (l <= 2.3e-195) {
		tmp = (2.0 / Math.pow((k / t_m), 2.0)) / (Math.pow(k, 2.0) * Math.pow((1.0 / (l / Math.pow(t_m, 1.5))), 2.0));
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(Math.sin(k), 2.0) * (Math.pow(k, 2.0) * t_m)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 2.3e-195:
		tmp = (2.0 / math.pow((k / t_m), 2.0)) / (math.pow(k, 2.0) * math.pow((1.0 / (l / math.pow(t_m, 1.5))), 2.0))
	else:
		tmp = (l * l) * (2.0 / ((math.pow(math.sin(k), 2.0) * (math.pow(k, 2.0) * t_m)) / math.cos(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 2.3e-195)
		tmp = Float64(Float64(2.0 / (Float64(k / t_m) ^ 2.0)) / Float64((k ^ 2.0) * (Float64(1.0 / Float64(l / (t_m ^ 1.5))) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64((k ^ 2.0) * t_m)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 2.3e-195)
		tmp = (2.0 / ((k / t_m) ^ 2.0)) / ((k ^ 2.0) * ((1.0 / (l / (t_m ^ 1.5))) ^ 2.0));
	else
		tmp = (l * l) * (2.0 / (((sin(k) ^ 2.0) * ((k ^ 2.0) * t_m)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[l, 2.3e-195], N[(N[(2.0 / N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[N[(1.0 / N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.3000000000000002e-195

   1. Initial program 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-/r* 31.1%

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

   3. Simplified 37.4%

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

      1. add-sqr-sqrt 14.6%

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

      2. pow2 14.6%

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

      3. sqrt-div 14.6%

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

      4. sqrt-pow1 19.9%

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

      5. metadata-eval 19.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 4.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 24.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 24.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. clear-num 24.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\color{blue}{\left(\frac{1}{\frac{\ell}{{t}^{1.5}}}\right)}}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. inv-pow 24.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\color{blue}{\left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}\right)}}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 24.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\color{blue}{\left({\left(\frac{\ell}{{t}^{1.5}}\right)}^{-1}\right)}}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. unpow-1 24.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\color{blue}{\left(\frac{1}{\frac{\ell}{{t}^{1.5}}}\right)}}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   10. Simplified 24.2%

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\color{blue}{\left(\frac{1}{\frac{\ell}{{t}^{1.5}}}\right)}}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   11. Taylor expanded in k around 0 22.2%

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{1}{\frac{\ell}{{t}^{1.5}}}\right)}^{2} \cdot \color{blue}{{k}^{2}}} \]

   if 2.3000000000000002e-195 < l

   1. Initial program 36.3%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. associate-*r* 79.8%

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

   7. Simplified 79.8%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.3 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot {\left(\frac{1}{\frac{\ell}{{t}^{1.5}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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) {
	return t_s * ((l * l) * (2.0 / ((Math.pow(Math.sin(k), 2.0) * (Math.pow(k, 2.0) * t_m)) / Math.cos(k))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 40.1%

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

5. Taylor expanded in t around 0 73.7%

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

   1. associate-*r* 73.7%

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

7. Simplified 73.7%

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

8. Final simplification 73.7%

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

Alternative 11: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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) {
	return t_s * ((l * l) * (Math.cos(k) * (2.0 / (Math.pow(Math.sin(k), 2.0) * (Math.pow(k, 2.0) * t_m)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

3. Simplified 40.1%

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

5. Taylor expanded in t around 0 73.7%

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

   1. associate-*r* 73.7%

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

7. Simplified 73.7%

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

   1. associate-/r/ 73.7%

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

   2. *-commutative 73.7%

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

9. Applied egg-rr 73.7%

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

10. Final simplification 73.7%

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

Alternative 12: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {k}^{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.000108:\\ \;\;\;\;-2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{-1}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{t\_2 \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (pow k 2.0) t_m)))
   (*
    t_s
    (if (<= k 0.000108)
      (* -2.0 (* (/ (pow l 2.0) (pow k 2.0)) (/ -1.0 t_2)))
      (*
       (* l l)
       (/ 2.0 (/ (* t_2 (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (cos k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(k, 2.0) * t_m;
	double tmp;
	if (k <= 0.000108) {
		tmp = -2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (-1.0 / t_2));
	} else {
		tmp = (l * l) * (2.0 / ((t_2 * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k ** 2.0d0) * t_m
    if (k <= 0.000108d0) then
        tmp = (-2.0d0) * (((l ** 2.0d0) / (k ** 2.0d0)) * ((-1.0d0) / t_2))
    else
        tmp = (l * l) * (2.0d0 / ((t_2 * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = Math.pow(k, 2.0) * t_m;
	double tmp;
	if (k <= 0.000108) {
		tmp = -2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (-1.0 / t_2));
	} else {
		tmp = (l * l) * (2.0 / ((t_2 * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(k, 2.0) * t_m
	tmp = 0
	if k <= 0.000108:
		tmp = -2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (-1.0 / t_2))
	else:
		tmp = (l * l) * (2.0 / ((t_2 * (0.5 - (math.cos((2.0 * k)) / 2.0))) / math.cos(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((k ^ 2.0) * t_m)
	tmp = 0.0
	if (k <= 0.000108)
		tmp = Float64(-2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(-1.0 / t_2)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(t_2 * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k ^ 2.0) * t_m;
	tmp = 0.0;
	if (k <= 0.000108)
		tmp = -2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (-1.0 / t_2));
	else
		tmp = (l * l) * (2.0 / ((t_2 * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 0.000108], N[(-2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(t$95$2 * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.08e-4

   1. Initial program 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-/r* 35.8%

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

   3. Simplified 40.1%

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

      1. add-sqr-sqrt 19.1%

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

      2. pow2 19.1%

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

      3. sqrt-div 19.1%

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

      4. sqrt-pow1 24.2%

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

      5. metadata-eval 24.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 16.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 28.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 28.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. times-frac 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 75.6%

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

   9. Simplified 75.6%

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

   10. Taylor expanded in k around 0 69.7%

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

   if 1.08e-4 < k

   1. Initial program 26.2%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in t around 0 69.9%

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

      1. associate-*r* 69.9%

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

   7. Simplified 69.9%

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

      1. unpow2 69.9%

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

      2. sin-mult 69.6%

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

   9. Applied egg-rr 69.6%

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

       1. div-sub 69.6%

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

       2. +-inverses 69.6%

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

       3. cos-0 69.6%

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

       4. metadata-eval 69.6%

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \cdot \left(\ell \cdot \ell\right) \]

       5. count-2 69.6%

          \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}{\cos k}} \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 69.6%

       \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}}{\cos k}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000108:\\ \;\;\;\;-2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{-1}{{k}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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) {
	return t_s * (-2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (-1.0 / (Math.pow(k, 2.0) * t_m))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

   2. associate-/r* 33.4%

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

3. Simplified 40.5%

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

   1. add-sqr-sqrt 18.6%

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

   2. pow2 18.6%

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

   3. sqrt-div 18.5%

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

   4. sqrt-pow1 23.1%

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

   5. metadata-eval 23.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. sqrt-prod 15.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. add-sqr-sqrt 27.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

6. Applied egg-rr 27.2%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in t around -inf 0.0%

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

   1. times-frac 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 74.6%

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

9. Simplified 74.6%

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

10. Taylor expanded in k around 0 66.8%

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

Alternative 14: 61.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 64.2%

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

   1. associate-/r* 64.2%

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

7. Simplified 64.2%

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

   1. associate-*l/ 65.0%

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

   2. div-inv 65.0%

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

   3. pow-flip 65.0%

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

   4. metadata-eval 65.0%

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

   5. pow2 65.0%

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

9. Applied egg-rr 65.0%

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

10. Final simplification 65.0%

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

Alternative 15: 61.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k, 4.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 64.2%

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

   1. *-commutative 64.2%

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

   2. associate-/r* 64.4%

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

7. Simplified 64.4%

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

Alternative 16: 61.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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) {
	return t_s * ((l * l) * (2.0 / (t_m * Math.pow(k, 4.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 64.2%

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

6. Final simplification 64.2%

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

Alternative 17: 61.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

3. Simplified 40.1%

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

5. Taylor expanded in k around 0 64.2%

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

   1. associate-/r* 64.2%

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

7. Simplified 64.2%

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

   1. div-inv 64.2%

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

   2. div-inv 64.2%

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

   3. pow-flip 64.2%

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

   4. metadata-eval 64.2%

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

9. Applied egg-rr 64.2%

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

    1. associate-*r/ 64.2%

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

    2. *-rgt-identity 64.2%

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

    3. *-commutative 64.2%

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

    4. associate-/l* 64.2%

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

11. Simplified 64.2%

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

12. Final simplification 64.2%

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

Reproduce

herbie shell --seed 2024186 
(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))))