Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.3% → 86.3%

Time: 18.3s

Alternatives: 14

Speedup: 2.0×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.3% 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: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{\sin k}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-277}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot t_2\right) \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+243}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2} \cdot {\ell}^{-2}}}{\frac{t_m}{\cos k \cdot {\sin k}^{-2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \left(t_2 \cdot \sqrt{\frac{\cos k}{t_m}}\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) (sin k))))
   (*
    t_s
    (if (<= (* l l) 2e-277)
      (pow (* (* (/ l k) t_2) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 2e+243)
        (/
         (/ 2.0 (* (pow k 2.0) (pow l -2.0)))
         (/ t_m (* (cos k) (pow (sin k) -2.0))))
        (pow (* (/ l k) (* t_2 (sqrt (/ (cos k) t_m)))) 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 t_2 = sqrt(2.0) / sin(k);
	double tmp;
	if ((l * l) <= 2e-277) {
		tmp = pow((((l / k) * t_2) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+243) {
		tmp = (2.0 / (pow(k, 2.0) * pow(l, -2.0))) / (t_m / (cos(k) * pow(sin(k), -2.0)));
	} else {
		tmp = pow(((l / k) * (t_2 * sqrt((cos(k) / t_m)))), 2.0);
	}
	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 = sqrt(2.0d0) / sin(k)
    if ((l * l) <= 2d-277) then
        tmp = (((l / k) * t_2) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 2d+243) then
        tmp = (2.0d0 / ((k ** 2.0d0) * (l ** (-2.0d0)))) / (t_m / (cos(k) * (sin(k) ** (-2.0d0))))
    else
        tmp = ((l / k) * (t_2 * sqrt((cos(k) / t_m)))) ** 2.0d0
    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.sqrt(2.0) / Math.sin(k);
	double tmp;
	if ((l * l) <= 2e-277) {
		tmp = Math.pow((((l / k) * t_2) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+243) {
		tmp = (2.0 / (Math.pow(k, 2.0) * Math.pow(l, -2.0))) / (t_m / (Math.cos(k) * Math.pow(Math.sin(k), -2.0)));
	} else {
		tmp = Math.pow(((l / k) * (t_2 * Math.sqrt((Math.cos(k) / t_m)))), 2.0);
	}
	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.sqrt(2.0) / math.sin(k)
	tmp = 0
	if (l * l) <= 2e-277:
		tmp = math.pow((((l / k) * t_2) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 2e+243:
		tmp = (2.0 / (math.pow(k, 2.0) * math.pow(l, -2.0))) / (t_m / (math.cos(k) * math.pow(math.sin(k), -2.0)))
	else:
		tmp = math.pow(((l / k) * (t_2 * math.sqrt((math.cos(k) / t_m)))), 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)
	t_2 = Float64(sqrt(2.0) / sin(k))
	tmp = 0.0
	if (Float64(l * l) <= 2e-277)
		tmp = Float64(Float64(Float64(l / k) * t_2) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 2e+243)
		tmp = Float64(Float64(2.0 / Float64((k ^ 2.0) * (l ^ -2.0))) / Float64(t_m / Float64(cos(k) * (sin(k) ^ -2.0))));
	else
		tmp = Float64(Float64(l / k) * Float64(t_2 * sqrt(Float64(cos(k) / t_m)))) ^ 2.0;
	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 = sqrt(2.0) / sin(k);
	tmp = 0.0;
	if ((l * l) <= 2e-277)
		tmp = (((l / k) * t_2) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 2e+243)
		tmp = (2.0 / ((k ^ 2.0) * (l ^ -2.0))) / (t_m / (cos(k) * (sin(k) ^ -2.0)));
	else
		tmp = ((l / k) * (t_2 * sqrt((cos(k) / t_m)))) ^ 2.0;
	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[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-277], N[Power[N[(N[(N[(l / k), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+243], N[(N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999994e-277

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. *-commutative 31.5%

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

      3. associate-*l* 31.5%

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

      4. associate-*l/ 31.5%

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

      5. +-commutative 31.5%

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

      6. unpow2 31.5%

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

      7. sqr-neg 31.5%

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

      8. distribute-frac-neg 31.5%

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

      9. distribute-frac-neg 31.5%

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

      10. unpow2 31.5%

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

      11. associate--l+ 44.7%

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

      12. metadata-eval 44.7%

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

      13. +-rgt-identity 44.7%

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

      14. unpow2 44.7%

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

      15. distribute-frac-neg 44.7%

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

      16. distribute-frac-neg 44.7%

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

   3. Simplified 44.7%

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

   5. Applied egg-rr 31.4%

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

   6. Taylor expanded in k around inf 54.0%

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

      1. times-frac 55.1%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in k around 0 44.4%

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

   if 1.99999999999999994e-277 < (*.f64 l l) < 2.0000000000000001e243

   1. Initial program 31.8%

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

   2. Taylor expanded in t around 0 89.4%

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

      1. times-frac 91.0%

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

      2. associate-/l* 91.0%

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

   4. Simplified 91.0%

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

      1. frac-times 88.8%

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

      2. div-inv 88.8%

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

      3. pow-flip 88.8%

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

      4. metadata-eval 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. times-frac 90.9%

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

   8. Simplified 90.9%

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

      1. expm1-log1p-u 65.5%

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

      2. expm1-udef 52.8%

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

      3. div-inv 52.8%

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

      4. pow-flip 52.8%

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

      5. metadata-eval 52.8%

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

   10. Applied egg-rr 52.8%

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

       1. expm1-def 65.5%

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

       2. expm1-log1p 90.9%

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

       3. associate-/r* 91.1%

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

   12. Simplified 91.1%

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

   if 2.0000000000000001e243 < (*.f64 l l)

   1. Initial program 39.5%

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

      1. associate-/r* 39.5%

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

      2. *-commutative 39.5%

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

      3. associate-*l* 39.5%

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

      4. associate-*l/ 39.5%

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

      5. +-commutative 39.5%

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

      6. unpow2 39.5%

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

      7. sqr-neg 39.5%

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

      8. distribute-frac-neg 39.5%

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

      9. distribute-frac-neg 39.5%

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

      10. unpow2 39.5%

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

      11. associate--l+ 39.5%

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

      12. metadata-eval 39.5%

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

      13. +-rgt-identity 39.5%

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

      14. unpow2 39.5%

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

      15. distribute-frac-neg 39.5%

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

      16. distribute-frac-neg 39.5%

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

   3. Simplified 39.5%

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

   5. Applied egg-rr 21.9%

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

   6. Taylor expanded in k around inf 36.0%

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

      1. times-frac 36.0%

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

   8. Simplified 36.0%

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

      1. pow1 36.0%

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

      2. associate-*l* 36.1%

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

   10. Applied egg-rr 36.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-277}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+243}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2} \cdot {\ell}^{-2}}}{\frac{t}{\cos k \cdot {\sin k}^{-2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \left(\frac{\sqrt{2}}{\sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)\right)}^{2}\\ \end{array} \]

Alternative 2: 79.6% accurate, 0.5× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (*
        (* (* (sin k) (/ (pow t_m 3.0) (* l l))) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) -1.0))
       INFINITY)
    (*
     (/ (/ 2.0 (sin k)) (/ (/ (pow t_m 3.0) l) (/ l (tan k))))
     (pow (/ k t_m) -2.0))
    (pow (* (* (/ l k) (/ (sqrt 2.0) (sin k))) (sqrt (/ 1.0 t_m))) 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 ((((sin(k) * (pow(t_m, 3.0) / (l * l))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + -1.0)) <= ((double) INFINITY)) {
		tmp = ((2.0 / sin(k)) / ((pow(t_m, 3.0) / l) / (l / tan(k)))) * pow((k / t_m), -2.0);
	} else {
		tmp = pow((((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))), 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 ((((Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + -1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 / Math.sin(k)) / ((Math.pow(t_m, 3.0) / l) / (l / Math.tan(k)))) * Math.pow((k / t_m), -2.0);
	} else {
		tmp = Math.pow((((l / k) * (Math.sqrt(2.0) / Math.sin(k))) * Math.sqrt((1.0 / t_m))), 2.0);
	}
	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 (((math.sin(k) * (math.pow(t_m, 3.0) / (l * l))) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + -1.0)) <= math.inf:
		tmp = ((2.0 / math.sin(k)) / ((math.pow(t_m, 3.0) / l) / (l / math.tan(k)))) * math.pow((k / t_m), -2.0)
	else:
		tmp = math.pow((((l / k) * (math.sqrt(2.0) / math.sin(k))) * math.sqrt((1.0 / t_m))), 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 (Float64(Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(2.0 / sin(k)) / Float64(Float64((t_m ^ 3.0) / l) / Float64(l / tan(k)))) * (Float64(k / t_m) ^ -2.0));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	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 ((((sin(k) * ((t_m ^ 3.0) / (l * l))) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + -1.0)) <= Inf)
		tmp = ((2.0 / sin(k)) / (((t_m ^ 3.0) / l) / (l / tan(k)))) * ((k / t_m) ^ -2.0);
	else
		tmp = (((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))) ^ 2.0;
	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[N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 87.4%

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

      1. associate-/r* 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 87.5%

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

      4. associate-*l/ 87.5%

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

      5. +-commutative 87.5%

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

      6. unpow2 87.5%

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

      7. sqr-neg 87.5%

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

      8. distribute-frac-neg 87.5%

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

      9. distribute-frac-neg 87.5%

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

      10. unpow2 87.5%

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

      11. associate--l+ 91.3%

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

      12. metadata-eval 91.3%

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

      13. +-rgt-identity 91.3%

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

      14. unpow2 91.3%

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

      15. distribute-frac-neg 91.3%

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

      16. distribute-frac-neg 91.3%

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

   3. Simplified 91.3%

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

      1. times-frac 95.1%

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

   5. Applied egg-rr 95.1%

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

      1. associate-*r/ 95.1%

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

   7. Applied egg-rr 95.1%

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

      1. pow2 95.1%

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

      2. div-inv 95.1%

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

      3. associate-/r* 95.1%

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

      4. associate-/l* 95.1%

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

      5. pow2 95.1%

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

      6. pow-flip 96.5%

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

      7. metadata-eval 96.5%

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

   9. Applied egg-rr 96.5%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

      4. associate-*l/ 0.0%

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

      5. +-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. sqr-neg 0.0%

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

      8. distribute-frac-neg 0.0%

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

      9. distribute-frac-neg 0.0%

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

      10. unpow2 0.0%

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

      11. associate--l+ 13.2%

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

      12. metadata-eval 13.2%

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

      13. +-rgt-identity 13.2%

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

      14. unpow2 13.2%

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

      15. distribute-frac-neg 13.2%

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

      16. distribute-frac-neg 13.2%

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

   3. Simplified 13.2%

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

   5. Applied egg-rr 20.0%

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

   6. Taylor expanded in k around inf 48.1%

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

      1. times-frac 48.7%

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

   8. Simplified 48.7%

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

   9. Taylor expanded in k around 0 31.2%

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

4. Final simplification 56.4%

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

Alternative 3: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-277}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+231}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t_m \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t_m}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-277)
    (pow (* (* (/ l k) (/ (sqrt 2.0) (sin k))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 5e+231)
      (*
       2.0
       (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0)))))
      (pow
       (* (sqrt (/ (cos k) t_m)) (/ (* l (sqrt 2.0)) (* k (sin 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 ((l * l) <= 2e-277) {
		tmp = pow((((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 5e+231) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	} else {
		tmp = pow((sqrt((cos(k) / t_m)) * ((l * sqrt(2.0)) / (k * sin(k)))), 2.0);
	}
	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 * l) <= 2d-277) then
        tmp = (((l / k) * (sqrt(2.0d0) / sin(k))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 5d+231) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (cos(k) / (t_m * (sin(k) ** 2.0d0))))
    else
        tmp = (sqrt((cos(k) / t_m)) * ((l * sqrt(2.0d0)) / (k * sin(k)))) ** 2.0d0
    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 * l) <= 2e-277) {
		tmp = Math.pow((((l / k) * (Math.sqrt(2.0) / Math.sin(k))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 5e+231) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = Math.pow((Math.sqrt((Math.cos(k) / t_m)) * ((l * Math.sqrt(2.0)) / (k * Math.sin(k)))), 2.0);
	}
	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 * l) <= 2e-277:
		tmp = math.pow((((l / k) * (math.sqrt(2.0) / math.sin(k))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 5e+231:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))))
	else:
		tmp = math.pow((math.sqrt((math.cos(k) / t_m)) * ((l * math.sqrt(2.0)) / (k * math.sin(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 (Float64(l * l) <= 2e-277)
		tmp = Float64(Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 5e+231)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(sqrt(Float64(cos(k) / t_m)) * Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k)))) ^ 2.0;
	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 * l) <= 2e-277)
		tmp = (((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 5e+231)
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / (t_m * (sin(k) ^ 2.0))));
	else
		tmp = (sqrt((cos(k) / t_m)) * ((l * sqrt(2.0)) / (k * sin(k)))) ^ 2.0;
	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[N[(l * l), $MachinePrecision], 2e-277], N[Power[N[(N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+231], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999994e-277

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. *-commutative 31.5%

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

      3. associate-*l* 31.5%

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

      4. associate-*l/ 31.5%

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

      5. +-commutative 31.5%

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

      6. unpow2 31.5%

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

      7. sqr-neg 31.5%

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

      8. distribute-frac-neg 31.5%

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

      9. distribute-frac-neg 31.5%

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

      10. unpow2 31.5%

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

      11. associate--l+ 44.7%

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

      12. metadata-eval 44.7%

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

      13. +-rgt-identity 44.7%

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

      14. unpow2 44.7%

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

      15. distribute-frac-neg 44.7%

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

      16. distribute-frac-neg 44.7%

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

   3. Simplified 44.7%

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

   5. Applied egg-rr 31.4%

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

   6. Taylor expanded in k around inf 54.0%

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

      1. times-frac 55.1%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in k around 0 44.4%

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

   if 1.99999999999999994e-277 < (*.f64 l l) < 5.00000000000000028e231

   1. Initial program 32.1%

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

      1. associate-/r* 32.3%

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

      2. *-commutative 32.3%

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

      3. associate-*l* 32.3%

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

      4. associate-*l/ 32.3%

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

      5. +-commutative 32.3%

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

      6. unpow2 32.3%

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

      7. sqr-neg 32.3%

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

      8. distribute-frac-neg 32.3%

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

      9. distribute-frac-neg 32.3%

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

      10. unpow2 32.3%

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

      11. associate--l+ 45.4%

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

      12. metadata-eval 45.4%

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

      13. +-rgt-identity 45.4%

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

      14. unpow2 45.4%

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

      15. distribute-frac-neg 45.4%

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

      16. distribute-frac-neg 45.4%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in k around inf 89.3%

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

      1. times-frac 91.0%

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

   6. Simplified 91.0%

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

   if 5.00000000000000028e231 < (*.f64 l l)

   1. Initial program 38.9%

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

      1. associate-/r* 38.9%

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

      2. *-commutative 38.9%

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

      3. associate-*l* 38.9%

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

      4. associate-*l/ 38.9%

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

      5. +-commutative 38.9%

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

      6. unpow2 38.9%

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

      7. sqr-neg 38.9%

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

      8. distribute-frac-neg 38.9%

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

      9. distribute-frac-neg 38.9%

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

      10. unpow2 38.9%

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

      11. associate--l+ 38.9%

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

      12. metadata-eval 38.9%

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

      13. +-rgt-identity 38.9%

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

      14. unpow2 38.9%

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

      15. distribute-frac-neg 38.9%

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

      16. distribute-frac-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Applied egg-rr 21.6%

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

   6. Taylor expanded in k around inf 36.9%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-277}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+231}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]

Alternative 4: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-277}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+243}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2} \cdot {\ell}^{-2}}}{\frac{t_m}{\cos k \cdot {\sin k}^{-2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t_m}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-277)
    (pow (* (* (/ l k) (/ (sqrt 2.0) (sin k))) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= (* l l) 2e+243)
      (/
       (/ 2.0 (* (pow k 2.0) (pow l -2.0)))
       (/ t_m (* (cos k) (pow (sin k) -2.0))))
      (pow
       (* (sqrt (/ (cos k) t_m)) (/ (* l (sqrt 2.0)) (* k (sin 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 ((l * l) <= 2e-277) {
		tmp = pow((((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+243) {
		tmp = (2.0 / (pow(k, 2.0) * pow(l, -2.0))) / (t_m / (cos(k) * pow(sin(k), -2.0)));
	} else {
		tmp = pow((sqrt((cos(k) / t_m)) * ((l * sqrt(2.0)) / (k * sin(k)))), 2.0);
	}
	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 * l) <= 2d-277) then
        tmp = (((l / k) * (sqrt(2.0d0) / sin(k))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 2d+243) then
        tmp = (2.0d0 / ((k ** 2.0d0) * (l ** (-2.0d0)))) / (t_m / (cos(k) * (sin(k) ** (-2.0d0))))
    else
        tmp = (sqrt((cos(k) / t_m)) * ((l * sqrt(2.0d0)) / (k * sin(k)))) ** 2.0d0
    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 * l) <= 2e-277) {
		tmp = Math.pow((((l / k) * (Math.sqrt(2.0) / Math.sin(k))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+243) {
		tmp = (2.0 / (Math.pow(k, 2.0) * Math.pow(l, -2.0))) / (t_m / (Math.cos(k) * Math.pow(Math.sin(k), -2.0)));
	} else {
		tmp = Math.pow((Math.sqrt((Math.cos(k) / t_m)) * ((l * Math.sqrt(2.0)) / (k * Math.sin(k)))), 2.0);
	}
	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 * l) <= 2e-277:
		tmp = math.pow((((l / k) * (math.sqrt(2.0) / math.sin(k))) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 2e+243:
		tmp = (2.0 / (math.pow(k, 2.0) * math.pow(l, -2.0))) / (t_m / (math.cos(k) * math.pow(math.sin(k), -2.0)))
	else:
		tmp = math.pow((math.sqrt((math.cos(k) / t_m)) * ((l * math.sqrt(2.0)) / (k * math.sin(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 (Float64(l * l) <= 2e-277)
		tmp = Float64(Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 2e+243)
		tmp = Float64(Float64(2.0 / Float64((k ^ 2.0) * (l ^ -2.0))) / Float64(t_m / Float64(cos(k) * (sin(k) ^ -2.0))));
	else
		tmp = Float64(sqrt(Float64(cos(k) / t_m)) * Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k)))) ^ 2.0;
	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 * l) <= 2e-277)
		tmp = (((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 2e+243)
		tmp = (2.0 / ((k ^ 2.0) * (l ^ -2.0))) / (t_m / (cos(k) * (sin(k) ^ -2.0)));
	else
		tmp = (sqrt((cos(k) / t_m)) * ((l * sqrt(2.0)) / (k * sin(k)))) ^ 2.0;
	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[N[(l * l), $MachinePrecision], 2e-277], N[Power[N[(N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+243], N[(N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.99999999999999994e-277

   1. Initial program 31.5%

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

      1. associate-/r* 31.5%

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

      2. *-commutative 31.5%

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

      3. associate-*l* 31.5%

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

      4. associate-*l/ 31.5%

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

      5. +-commutative 31.5%

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

      6. unpow2 31.5%

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

      7. sqr-neg 31.5%

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

      8. distribute-frac-neg 31.5%

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

      9. distribute-frac-neg 31.5%

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

      10. unpow2 31.5%

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

      11. associate--l+ 44.7%

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

      12. metadata-eval 44.7%

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

      13. +-rgt-identity 44.7%

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

      14. unpow2 44.7%

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

      15. distribute-frac-neg 44.7%

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

      16. distribute-frac-neg 44.7%

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

   3. Simplified 44.7%

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

   5. Applied egg-rr 31.4%

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

   6. Taylor expanded in k around inf 54.0%

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

      1. times-frac 55.1%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in k around 0 44.4%

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

   if 1.99999999999999994e-277 < (*.f64 l l) < 2.0000000000000001e243

   1. Initial program 31.8%

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

   2. Taylor expanded in t around 0 89.4%

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

      1. times-frac 91.0%

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

      2. associate-/l* 91.0%

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

   4. Simplified 91.0%

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

      1. frac-times 88.8%

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

      2. div-inv 88.8%

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

      3. pow-flip 88.8%

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

      4. metadata-eval 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. times-frac 90.9%

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

   8. Simplified 90.9%

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

      1. expm1-log1p-u 65.5%

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

      2. expm1-udef 52.8%

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

      3. div-inv 52.8%

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

      4. pow-flip 52.8%

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

      5. metadata-eval 52.8%

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

   10. Applied egg-rr 52.8%

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

       1. expm1-def 65.5%

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

       2. expm1-log1p 90.9%

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

       3. associate-/r* 91.1%

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

   12. Simplified 91.1%

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

   if 2.0000000000000001e243 < (*.f64 l l)

   1. Initial program 39.5%

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

      1. associate-/r* 39.5%

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

      2. *-commutative 39.5%

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

      3. associate-*l* 39.5%

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

      4. associate-*l/ 39.5%

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

      5. +-commutative 39.5%

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

      6. unpow2 39.5%

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

      7. sqr-neg 39.5%

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

      8. distribute-frac-neg 39.5%

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

      9. distribute-frac-neg 39.5%

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

      10. unpow2 39.5%

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

      11. associate--l+ 39.5%

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

      12. metadata-eval 39.5%

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

      13. +-rgt-identity 39.5%

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

      14. unpow2 39.5%

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

      15. distribute-frac-neg 39.5%

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

      16. distribute-frac-neg 39.5%

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

   3. Simplified 39.5%

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

   5. Applied egg-rr 21.9%

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

   6. Taylor expanded in k around inf 36.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-277}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+243}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2} \cdot {\ell}^{-2}}}{\frac{t}{\cos k \cdot {\sin k}^{-2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]

Alternative 5: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-12}:\\ \;\;\;\;{\left(t_2 \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(t_2 \cdot \sqrt{\frac{\cos k}{t_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ l k) (/ (sqrt 2.0) (sin k)))))
   (*
    t_s
    (if (<= (* l l) 2e-12)
      (pow (* t_2 (sqrt (/ 1.0 t_m))) 2.0)
      (pow (* t_2 (sqrt (/ (cos k) t_m))) 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 t_2 = (l / k) * (sqrt(2.0) / sin(k));
	double tmp;
	if ((l * l) <= 2e-12) {
		tmp = pow((t_2 * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = pow((t_2 * sqrt((cos(k) / t_m))), 2.0);
	}
	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 = (l / k) * (sqrt(2.0d0) / sin(k))
    if ((l * l) <= 2d-12) then
        tmp = (t_2 * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (t_2 * sqrt((cos(k) / t_m))) ** 2.0d0
    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 = (l / k) * (Math.sqrt(2.0) / Math.sin(k));
	double tmp;
	if ((l * l) <= 2e-12) {
		tmp = Math.pow((t_2 * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = Math.pow((t_2 * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	}
	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 = (l / k) * (math.sqrt(2.0) / math.sin(k))
	tmp = 0
	if (l * l) <= 2e-12:
		tmp = math.pow((t_2 * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = math.pow((t_2 * math.sqrt((math.cos(k) / t_m))), 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)
	t_2 = Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k)))
	tmp = 0.0
	if (Float64(l * l) <= 2e-12)
		tmp = Float64(t_2 * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(t_2 * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	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 = (l / k) * (sqrt(2.0) / sin(k));
	tmp = 0.0;
	if ((l * l) <= 2e-12)
		tmp = (t_2 * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (t_2 * sqrt((cos(k) / t_m))) ^ 2.0;
	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[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-12], N[Power[N[(t$95$2 * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(t$95$2 * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999996e-12

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-*l* 32.4%

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

      4. associate-*l/ 32.4%

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

      5. +-commutative 32.4%

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

      6. unpow2 32.4%

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

      7. sqr-neg 32.4%

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

      8. distribute-frac-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. unpow2 32.4%

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

      11. associate--l+ 45.4%

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

      12. metadata-eval 45.4%

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

      13. +-rgt-identity 45.4%

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

      14. unpow2 45.4%

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

      15. distribute-frac-neg 45.4%

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

      16. distribute-frac-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in k around inf 50.6%

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

      1. times-frac 51.3%

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

   8. Simplified 51.3%

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

   9. Taylor expanded in k around 0 40.7%

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

   if 1.99999999999999996e-12 < (*.f64 l l)

   1. Initial program 35.2%

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

      1. associate-/r* 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-*l* 35.3%

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

      4. associate-*l/ 35.3%

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

      5. +-commutative 35.3%

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

      6. unpow2 35.3%

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

      7. sqr-neg 35.3%

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

      8. distribute-frac-neg 35.3%

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

      9. distribute-frac-neg 35.3%

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

      10. unpow2 35.3%

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

      11. associate--l+ 41.3%

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

      12. metadata-eval 41.3%

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

      13. +-rgt-identity 41.3%

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

      14. unpow2 41.3%

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

      15. distribute-frac-neg 41.3%

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

      16. distribute-frac-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in k around inf 44.0%

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

      1. times-frac 44.0%

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

   8. Simplified 44.0%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-12}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \end{array} \]

Alternative 6: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-12}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t_m}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e-12)
    (pow (* (* (/ l k) (/ (sqrt 2.0) (sin k))) (sqrt (/ 1.0 t_m))) 2.0)
    (pow (* (sqrt (/ (cos k) t_m)) (/ (* l (sqrt 2.0)) (* k (sin 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 ((l * l) <= 2e-12) {
		tmp = pow((((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = pow((sqrt((cos(k) / t_m)) * ((l * sqrt(2.0)) / (k * sin(k)))), 2.0);
	}
	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 * l) <= 2d-12) then
        tmp = (((l / k) * (sqrt(2.0d0) / sin(k))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (sqrt((cos(k) / t_m)) * ((l * sqrt(2.0d0)) / (k * sin(k)))) ** 2.0d0
    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 * l) <= 2e-12) {
		tmp = Math.pow((((l / k) * (Math.sqrt(2.0) / Math.sin(k))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = Math.pow((Math.sqrt((Math.cos(k) / t_m)) * ((l * Math.sqrt(2.0)) / (k * Math.sin(k)))), 2.0);
	}
	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 * l) <= 2e-12:
		tmp = math.pow((((l / k) * (math.sqrt(2.0) / math.sin(k))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = math.pow((math.sqrt((math.cos(k) / t_m)) * ((l * math.sqrt(2.0)) / (k * math.sin(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 (Float64(l * l) <= 2e-12)
		tmp = Float64(Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(sqrt(Float64(cos(k) / t_m)) * Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k)))) ^ 2.0;
	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 * l) <= 2e-12)
		tmp = (((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (sqrt((cos(k) / t_m)) * ((l * sqrt(2.0)) / (k * sin(k)))) ^ 2.0;
	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[N[(l * l), $MachinePrecision], 2e-12], N[Power[N[(N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999996e-12

   1. Initial program 32.4%

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

      1. associate-/r* 32.4%

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

      2. *-commutative 32.4%

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

      3. associate-*l* 32.4%

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

      4. associate-*l/ 32.4%

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

      5. +-commutative 32.4%

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

      6. unpow2 32.4%

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

      7. sqr-neg 32.4%

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

      8. distribute-frac-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. unpow2 32.4%

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

      11. associate--l+ 45.4%

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

      12. metadata-eval 45.4%

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

      13. +-rgt-identity 45.4%

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

      14. unpow2 45.4%

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

      15. distribute-frac-neg 45.4%

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

      16. distribute-frac-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in k around inf 50.6%

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

      1. times-frac 51.3%

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

   8. Simplified 51.3%

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

   9. Taylor expanded in k around 0 40.7%

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

   if 1.99999999999999996e-12 < (*.f64 l l)

   1. Initial program 35.2%

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

      1. associate-/r* 35.3%

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

      2. *-commutative 35.3%

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

      3. associate-*l* 35.3%

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

      4. associate-*l/ 35.3%

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

      5. +-commutative 35.3%

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

      6. unpow2 35.3%

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

      7. sqr-neg 35.3%

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

      8. distribute-frac-neg 35.3%

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

      9. distribute-frac-neg 35.3%

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

      10. unpow2 35.3%

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

      11. associate--l+ 41.3%

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

      12. metadata-eval 41.3%

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

      13. +-rgt-identity 41.3%

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

      14. unpow2 41.3%

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

      15. distribute-frac-neg 41.3%

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

      16. distribute-frac-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Applied egg-rr 26.2%

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

   6. Taylor expanded in k around inf 44.0%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-12}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (pow (* (* (/ l k) (/ (sqrt 2.0) (sin k))) (sqrt (/ 1.0 t_m))) 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) {
	return t_s * pow((((l / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))), 2.0);
}

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 / k) * (sqrt(2.0d0) / sin(k))) * sqrt((1.0d0 / t_m))) ** 2.0d0)
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 * Math.pow((((l / k) * (Math.sqrt(2.0) / Math.sin(k))) * Math.sqrt((1.0 / t_m))), 2.0);
}

Python

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

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(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) * sqrt(Float64(1.0 / t_m))) ^ 2.0))
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 / k) * (sqrt(2.0) / sin(k))) * sqrt((1.0 / t_m))) ^ 2.0);
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[Power[N[(N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. associate-/r* 33.8%

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

   2. *-commutative 33.8%

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

   3. associate-*l* 33.8%

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

   4. associate-*l/ 33.8%

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

   5. +-commutative 33.8%

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

   6. unpow2 33.8%

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

   7. sqr-neg 33.8%

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

   8. distribute-frac-neg 33.8%

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

   9. distribute-frac-neg 33.8%

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

   10. unpow2 33.8%

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

   11. associate--l+ 43.4%

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

   12. metadata-eval 43.4%

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

   13. +-rgt-identity 43.4%

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

   14. unpow2 43.4%

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

   15. distribute-frac-neg 43.4%

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

   16. distribute-frac-neg 43.4%

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

3. Simplified 43.4%

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

5. Applied egg-rr 26.7%

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

6. Taylor expanded in k around inf 47.4%

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

   1. times-frac 47.7%

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

8. Simplified 47.7%

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

9. Taylor expanded in k around 0 34.3%

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

10. Final simplification 34.3%

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

Alternative 8: 71.9% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (pow (* (sqrt (/ 1.0 t_m)) (/ l (/ (pow k 2.0) (sqrt 2.0)))) 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) {
	return t_s * pow((sqrt((1.0 / t_m)) * (l / (pow(k, 2.0) / sqrt(2.0)))), 2.0);
}

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 * ((sqrt((1.0d0 / t_m)) * (l / ((k ** 2.0d0) / sqrt(2.0d0)))) ** 2.0d0)
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 * Math.pow((Math.sqrt((1.0 / t_m)) * (l / (Math.pow(k, 2.0) / Math.sqrt(2.0)))), 2.0);
}

Python

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

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(sqrt(Float64(1.0 / t_m)) * Float64(l / Float64((k ^ 2.0) / sqrt(2.0)))) ^ 2.0))
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 * ((sqrt((1.0 / t_m)) * (l / ((k ^ 2.0) / sqrt(2.0)))) ^ 2.0);
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[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. associate-/r* 33.8%

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

   2. *-commutative 33.8%

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

   3. associate-*l* 33.8%

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

   4. associate-*l/ 33.8%

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

   5. +-commutative 33.8%

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

   6. unpow2 33.8%

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

   7. sqr-neg 33.8%

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

   8. distribute-frac-neg 33.8%

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

   9. distribute-frac-neg 33.8%

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

   10. unpow2 33.8%

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

   11. associate--l+ 43.4%

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

   12. metadata-eval 43.4%

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

   13. +-rgt-identity 43.4%

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

   14. unpow2 43.4%

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

   15. distribute-frac-neg 43.4%

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

   16. distribute-frac-neg 43.4%

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

3. Simplified 43.4%

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

5. Applied egg-rr 26.7%

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

6. Taylor expanded in k around 0 33.1%

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

   1. associate-/l* 33.1%

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

8. Simplified 33.1%

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

9. Final simplification 33.1%

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

Alternative 9: 71.9% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (pow (* (sqrt (/ 1.0 t_m)) (/ (* l (sqrt 2.0)) (pow k 2.0))) 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) {
	return t_s * pow((sqrt((1.0 / t_m)) * ((l * sqrt(2.0)) / pow(k, 2.0))), 2.0);
}

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 * ((sqrt((1.0d0 / t_m)) * ((l * sqrt(2.0d0)) / (k ** 2.0d0))) ** 2.0d0)
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 * Math.pow((Math.sqrt((1.0 / t_m)) * ((l * Math.sqrt(2.0)) / Math.pow(k, 2.0))), 2.0);
}

Python

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

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(sqrt(Float64(1.0 / t_m)) * Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0))) ^ 2.0))
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 * ((sqrt((1.0 / t_m)) * ((l * sqrt(2.0)) / (k ^ 2.0))) ^ 2.0);
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[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

   1. associate-/r* 33.8%

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

   2. *-commutative 33.8%

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

   3. associate-*l* 33.8%

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

   4. associate-*l/ 33.8%

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

   5. +-commutative 33.8%

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

   6. unpow2 33.8%

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

   7. sqr-neg 33.8%

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

   8. distribute-frac-neg 33.8%

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

   9. distribute-frac-neg 33.8%

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

   10. unpow2 33.8%

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

   11. associate--l+ 43.4%

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

   12. metadata-eval 43.4%

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

   13. +-rgt-identity 43.4%

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

   14. unpow2 43.4%

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

   15. distribute-frac-neg 43.4%

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

   16. distribute-frac-neg 43.4%

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

3. Simplified 43.4%

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

5. Applied egg-rr 26.7%

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

6. Taylor expanded in k around 0 33.1%

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

7. Final simplification 33.1%

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

Alternative 10: 68.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8.8 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t_m}\\ \mathbf{elif}\;t_m \leq 0.002:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t_m}}}{k \cdot \frac{{t_m}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right)\right) \cdot \left(\frac{k}{t_m} \cdot \frac{k}{t_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t_m \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.8e-212)
    (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t_m))
    (if (<= t_m 0.002)
      (pow (/ (/ (sqrt 2.0) (/ k t_m)) (* k (/ (pow t_m 1.5) l))) 2.0)
      (if (<= t_m 5.5e+102)
        (/
         2.0
         (*
          (* (tan k) (* (sin k) (/ (/ (pow t_m 3.0) l) l)))
          (* (/ k t_m) (/ k t_m))))
        (/ 2.0 (* (/ (pow k 2.0) (pow l 2.0)) (* t_m (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 (t_m <= 8.8e-212) {
		tmp = 2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / t_m);
	} else if (t_m <= 0.002) {
		tmp = pow(((sqrt(2.0) / (k / t_m)) / (k * (pow(t_m, 1.5) / l))), 2.0);
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((pow(t_m, 3.0) / l) / l))) * ((k / t_m) * (k / t_m)));
	} else {
		tmp = 2.0 / ((pow(k, 2.0) / pow(l, 2.0)) * (t_m * pow(k, 2.0)));
	}
	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 (t_m <= 8.8d-212) then
        tmp = 2.0d0 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t_m)
    else if (t_m <= 0.002d0) then
        tmp = ((sqrt(2.0d0) / (k / t_m)) / (k * ((t_m ** 1.5d0) / l))) ** 2.0d0
    else if (t_m <= 5.5d+102) then
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 3.0d0) / l) / l))) * ((k / t_m) * (k / t_m)))
    else
        tmp = 2.0d0 / (((k ** 2.0d0) / (l ** 2.0d0)) * (t_m * (k ** 2.0d0)))
    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 (t_m <= 8.8e-212) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.pow(k, -4.0)) / t_m);
	} else if (t_m <= 0.002) {
		tmp = Math.pow(((Math.sqrt(2.0) / (k / t_m)) / (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l))) * ((k / t_m) * (k / t_m)));
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * (t_m * Math.pow(k, 2.0)));
	}
	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 t_m <= 8.8e-212:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.pow(k, -4.0)) / t_m)
	elif t_m <= 0.002:
		tmp = math.pow(((math.sqrt(2.0) / (k / t_m)) / (k * (math.pow(t_m, 1.5) / l))), 2.0)
	elif t_m <= 5.5e+102:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * ((math.pow(t_m, 3.0) / l) / l))) * ((k / t_m) * (k / t_m)))
	else:
		tmp = 2.0 / ((math.pow(k, 2.0) / math.pow(l, 2.0)) * (t_m * 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 (t_m <= 8.8e-212)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * (k ^ -4.0)) / t_m));
	elseif (t_m <= 0.002)
		tmp = Float64(Float64(sqrt(2.0) / Float64(k / t_m)) / Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0;
	elseif (t_m <= 5.5e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))) * Float64(Float64(k / t_m) * Float64(k / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(t_m * (k ^ 2.0))));
	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 (t_m <= 8.8e-212)
		tmp = 2.0 * (((l ^ 2.0) * (k ^ -4.0)) / t_m);
	elseif (t_m <= 0.002)
		tmp = ((sqrt(2.0) / (k / t_m)) / (k * ((t_m ^ 1.5) / l))) ^ 2.0;
	elseif (t_m <= 5.5e+102)
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 3.0) / l) / l))) * ((k / t_m) * (k / t_m)));
	else
		tmp = 2.0 / (((k ^ 2.0) / (l ^ 2.0)) * (t_m * (k ^ 2.0)));
	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[t$95$m, 8.8e-212], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.002], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 5.5e+102], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 8.80000000000000012e-212

   1. Initial program 32.8%

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

      1. associate-/r* 32.8%

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

      2. *-commutative 32.8%

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

      3. associate-*l* 32.8%

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

      4. associate-*l/ 32.8%

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

      5. +-commutative 32.8%

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

      6. unpow2 32.8%

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

      7. sqr-neg 32.8%

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

      8. distribute-frac-neg 32.8%

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

      9. distribute-frac-neg 32.8%

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

      10. unpow2 32.8%

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

      11. associate--l+ 39.6%

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

      12. metadata-eval 39.6%

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

      13. +-rgt-identity 39.6%

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

      14. unpow2 39.6%

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

      15. distribute-frac-neg 39.6%

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

      16. distribute-frac-neg 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in k around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-/r* 61.2%

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

   6. Simplified 61.2%

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

      1. div-inv 61.2%

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

      2. pow-flip 61.8%

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

      3. metadata-eval 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. associate-*l/ 62.7%

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

   10. Applied egg-rr 62.7%

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

   if 8.80000000000000012e-212 < t < 2e-3

   1. Initial program 44.2%

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

      1. associate-/r* 44.2%

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

      2. *-commutative 44.2%

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

      3. associate-*l* 44.2%

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

      4. associate-*l/ 44.2%

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

      5. +-commutative 44.2%

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

      6. unpow2 44.2%

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

      7. sqr-neg 44.2%

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

      8. distribute-frac-neg 44.2%

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

      9. distribute-frac-neg 44.2%

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

      10. unpow2 44.2%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

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

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

   3. Simplified 48.3%

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

   5. Applied egg-rr 68.9%

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

   6. Taylor expanded in k around 0 77.6%

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

   if 2e-3 < t < 5.49999999999999981e102

   1. Initial program 50.5%

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

      1. associate-/r* 57.4%

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

      2. div-inv 57.3%

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

   3. Applied egg-rr 57.3%

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

      1. +-commutative 57.3%

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

      2. associate--l+ 84.2%

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

      3. metadata-eval 84.2%

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

      4. +-rgt-identity 84.2%

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

      5. unpow2 84.2%

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

   5. Applied egg-rr 84.2%

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

      1. un-div-inv 84.5%

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

   7. Applied egg-rr 84.5%

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

   if 5.49999999999999981e102 < t

   1. Initial program 18.4%

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

   2. Taylor expanded in t around 0 76.6%

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

      1. times-frac 82.8%

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

      2. associate-/l* 82.8%

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

   4. Simplified 82.8%

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

      1. frac-times 74.3%

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

      2. div-inv 74.3%

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

      3. pow-flip 74.3%

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

      4. metadata-eval 74.3%

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

   6. Applied egg-rr 74.3%

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

      1. times-frac 82.8%

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

   8. Simplified 82.8%

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

   9. Taylor expanded in k around 0 75.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t}\\ \mathbf{elif}\;t \leq 0.002:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{k \cdot \frac{{t}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 11: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.8 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t_m}\\ \mathbf{elif}\;t_m \leq 1.9 \cdot 10^{+183}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t_m}}}{k \cdot \frac{{t_m}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t_m \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-212)
    (* 2.0 (/ (* (pow l 2.0) (pow k -4.0)) t_m))
    (if (<= t_m 1.9e+183)
      (pow (/ (/ (sqrt 2.0) (/ k t_m)) (* k (/ (pow t_m 1.5) l))) 2.0)
      (/ 2.0 (* (/ (pow k 2.0) (pow l 2.0)) (* t_m (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 (t_m <= 2.8e-212) {
		tmp = 2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / t_m);
	} else if (t_m <= 1.9e+183) {
		tmp = pow(((sqrt(2.0) / (k / t_m)) / (k * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) / pow(l, 2.0)) * (t_m * pow(k, 2.0)));
	}
	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 (t_m <= 2.8d-212) then
        tmp = 2.0d0 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t_m)
    else if (t_m <= 1.9d+183) then
        tmp = ((sqrt(2.0d0) / (k / t_m)) / (k * ((t_m ** 1.5d0) / l))) ** 2.0d0
    else
        tmp = 2.0d0 / (((k ** 2.0d0) / (l ** 2.0d0)) * (t_m * (k ** 2.0d0)))
    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 (t_m <= 2.8e-212) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.pow(k, -4.0)) / t_m);
	} else if (t_m <= 1.9e+183) {
		tmp = Math.pow(((Math.sqrt(2.0) / (k / t_m)) / (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * (t_m * Math.pow(k, 2.0)));
	}
	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 t_m <= 2.8e-212:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.pow(k, -4.0)) / t_m)
	elif t_m <= 1.9e+183:
		tmp = math.pow(((math.sqrt(2.0) / (k / t_m)) / (k * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = 2.0 / ((math.pow(k, 2.0) / math.pow(l, 2.0)) * (t_m * 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 (t_m <= 2.8e-212)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * (k ^ -4.0)) / t_m));
	elseif (t_m <= 1.9e+183)
		tmp = Float64(Float64(sqrt(2.0) / Float64(k / t_m)) / Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(t_m * (k ^ 2.0))));
	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 (t_m <= 2.8e-212)
		tmp = 2.0 * (((l ^ 2.0) * (k ^ -4.0)) / t_m);
	elseif (t_m <= 1.9e+183)
		tmp = ((sqrt(2.0) / (k / t_m)) / (k * ((t_m ^ 1.5) / l))) ^ 2.0;
	else
		tmp = 2.0 / (((k ^ 2.0) / (l ^ 2.0)) * (t_m * (k ^ 2.0)));
	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[t$95$m, 2.8e-212], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+183], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.80000000000000014e-212

   1. Initial program 32.8%

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

      1. associate-/r* 32.8%

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

      2. *-commutative 32.8%

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

      3. associate-*l* 32.8%

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

      4. associate-*l/ 32.8%

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

      5. +-commutative 32.8%

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

      6. unpow2 32.8%

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

      7. sqr-neg 32.8%

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

      8. distribute-frac-neg 32.8%

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

      9. distribute-frac-neg 32.8%

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

      10. unpow2 32.8%

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

      11. associate--l+ 39.6%

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

      12. metadata-eval 39.6%

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

      13. +-rgt-identity 39.6%

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

      14. unpow2 39.6%

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

      15. distribute-frac-neg 39.6%

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

      16. distribute-frac-neg 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in k around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-/r* 61.2%

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

   6. Simplified 61.2%

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

      1. div-inv 61.2%

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

      2. pow-flip 61.8%

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

      3. metadata-eval 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. associate-*l/ 62.7%

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

   10. Applied egg-rr 62.7%

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

   if 2.80000000000000014e-212 < t < 1.9e183

   1. Initial program 42.9%

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

      1. associate-/r* 43.0%

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

      2. *-commutative 43.0%

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

      3. associate-*l* 43.0%

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

      4. associate-*l/ 43.0%

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

      5. +-commutative 43.0%

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

      6. unpow2 43.0%

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

      7. sqr-neg 43.0%

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

      8. distribute-frac-neg 43.0%

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

      9. distribute-frac-neg 43.0%

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

      10. unpow2 43.0%

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

      11. associate--l+ 51.6%

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

      12. metadata-eval 51.6%

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

      13. +-rgt-identity 51.6%

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

      14. unpow2 51.6%

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

      15. distribute-frac-neg 51.6%

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

      16. distribute-frac-neg 51.6%

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

   3. Simplified 51.6%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in k around 0 72.0%

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

   if 1.9e183 < t

   1. Initial program 5.0%

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

   2. Taylor expanded in t around 0 80.2%

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

      1. times-frac 86.8%

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

      2. associate-/l* 86.8%

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

   4. Simplified 86.8%

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

      1. frac-times 75.8%

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

      2. div-inv 75.8%

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

      3. pow-flip 75.9%

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

      4. metadata-eval 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. times-frac 86.9%

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

   8. Simplified 86.9%

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

   9. Taylor expanded in k around 0 77.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+183}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{k \cdot \frac{{t}^{1.5}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 12: 63.4% accurate, 1.3× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (/ (pow k 2.0) (pow l 2.0)) (* t_m (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) {
	return t_s * (2.0 / ((pow(k, 2.0) / pow(l, 2.0)) * (t_m * pow(k, 2.0))));
}

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 / (((k ** 2.0d0) / (l ** 2.0d0)) * (t_m * (k ** 2.0d0))))
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(k, 2.0) / Math.pow(l, 2.0)) * (t_m * Math.pow(k, 2.0))));
}

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(k, 2.0) / math.pow(l, 2.0)) * (t_m * math.pow(k, 2.0))))

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((k ^ 2.0) / (l ^ 2.0)) * Float64(t_m * (k ^ 2.0)))))
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 / (((k ^ 2.0) / (l ^ 2.0)) * (t_m * (k ^ 2.0))));
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[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.8%

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

2. Taylor expanded in t around 0 73.4%

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

   1. times-frac 73.9%

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

   2. associate-/l* 73.9%

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

4. Simplified 73.9%

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

   1. frac-times 73.2%

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

   2. div-inv 73.2%

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

   3. pow-flip 73.2%

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

   4. metadata-eval 73.2%

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

6. Applied egg-rr 73.2%

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

   1. times-frac 73.8%

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

8. Simplified 73.8%

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

9. Taylor expanded in k around 0 64.4%

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

10. Final simplification 64.4%

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

Alternative 13: 60.1% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 33.8%

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

   1. associate-/r* 33.8%

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

   2. *-commutative 33.8%

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

   3. associate-*l* 33.8%

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

   4. associate-*l/ 33.8%

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

   5. +-commutative 33.8%

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

   6. unpow2 33.8%

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

   7. sqr-neg 33.8%

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

   8. distribute-frac-neg 33.8%

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

   9. distribute-frac-neg 33.8%

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

   10. unpow2 33.8%

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

   11. associate--l+ 43.4%

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

   12. metadata-eval 43.4%

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

   13. +-rgt-identity 43.4%

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

   14. unpow2 43.4%

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

   15. distribute-frac-neg 43.4%

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

   16. distribute-frac-neg 43.4%

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

3. Simplified 43.4%

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

4. Taylor expanded in k around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-/r* 60.2%

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

6. Simplified 60.2%

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

   1. div-inv 60.2%

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

   2. pow-flip 60.6%

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

   3. metadata-eval 60.6%

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

8. Applied egg-rr 60.6%

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

9. Final simplification 60.6%

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

Alternative 14: 61.0% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 33.8%

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

   1. associate-/r* 33.8%

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

   2. *-commutative 33.8%

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

   3. associate-*l* 33.8%

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

   4. associate-*l/ 33.8%

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

   5. +-commutative 33.8%

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

   6. unpow2 33.8%

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

   7. sqr-neg 33.8%

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

   8. distribute-frac-neg 33.8%

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

   9. distribute-frac-neg 33.8%

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

   10. unpow2 33.8%

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

   11. associate--l+ 43.4%

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

   12. metadata-eval 43.4%

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

   13. +-rgt-identity 43.4%

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

   14. unpow2 43.4%

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

   15. distribute-frac-neg 43.4%

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

   16. distribute-frac-neg 43.4%

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

3. Simplified 43.4%

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

4. Taylor expanded in k around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-/r* 60.2%

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

6. Simplified 60.2%

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

   1. div-inv 60.2%

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

   2. pow-flip 60.6%

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

   3. metadata-eval 60.6%

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

8. Applied egg-rr 60.6%

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

   1. associate-*l/ 62.7%

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

10. Applied egg-rr 62.7%

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

11. Final simplification 62.7%

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

Reproduce

herbie shell --seed 2023339 
(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))))