Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 99.2%

Time: 34.7s

Alternatives: 12

Speedup: 3.8×

• 39.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 6.6e-30)
    (/ 2.0 (pow (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))) 2.0))
    (/ 2.0 (* (* (sin k_m) (tan k_m)) (pow (* (/ k_m l) (sqrt t_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-30) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))), 2.0);
	} else {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * pow(((k_m / l) * sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6.6d-30) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((sin(k_m) * tan(k_m)) * (((k_m / l) * sqrt(t_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-30) {
		tmp = 2.0 / Math.pow(((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m)))), 2.0);
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(((k_m / l) * Math.sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 6.6e-30:
		tmp = 2.0 / math.pow(((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m)))), 2.0)
	else:
		tmp = 2.0 / ((math.sin(k_m) * math.tan(k_m)) * math.pow(((k_m / l) * math.sqrt(t_m)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6e-30)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(Float64(k_m / l) * sqrt(t_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 6.6e-30)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m)))) ^ 2.0);
	else
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * (((k_m / l) * sqrt(t_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 6.6e-30], N[(2.0 / N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.6000000000000006e-30

   1. Initial program 35.0%

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

   4. Applied egg-rr 33.8%

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

      1. associate-*r/ 33.8%

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

      2. metadata-eval 33.8%

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

      3. associate-*r* 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in k around inf 52.1%

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

      1. associate-/l* 53.7%

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

   9. Simplified 53.7%

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

   if 6.6000000000000006e-30 < k

   1. Initial program 24.5%

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

   4. Applied egg-rr 23.3%

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

      1. associate-*r/ 23.3%

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

      2. metadata-eval 23.3%

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

      3. associate-*r* 23.3%

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

   6. Simplified 23.3%

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

      1. *-un-lft-identity 23.3%

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

      2. *-commutative 23.3%

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

      3. unpow-prod-down 23.4%

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

      4. pow2 23.4%

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

      5. add-sqr-sqrt 36.5%

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

   8. Applied egg-rr 36.5%

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

      1. *-lft-identity 36.5%

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

      2. times-frac 36.3%

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

      3. *-commutative 36.3%

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

      4. times-frac 44.9%

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

   10. Simplified 44.9%

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

   11. Taylor expanded in t around 0 53.4%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot {\left(k\_m \cdot \frac{\sin k\_m \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}{\ell}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.8e-29)
    (* 2.0 (pow (* k_m (/ (* (sin k_m) (sqrt (/ t_m (cos k_m)))) l)) -2.0))
    (/ 2.0 (* (* (sin k_m) (tan k_m)) (pow (* (/ k_m l) (sqrt t_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-29) {
		tmp = 2.0 * pow((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) / l)), -2.0);
	} else {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * pow(((k_m / l) * sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d-29) then
        tmp = 2.0d0 * ((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) / l)) ** (-2.0d0))
    else
        tmp = 2.0d0 / ((sin(k_m) * tan(k_m)) * (((k_m / l) * sqrt(t_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-29) {
		tmp = 2.0 * Math.pow((k_m * ((Math.sin(k_m) * Math.sqrt((t_m / Math.cos(k_m)))) / l)), -2.0);
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(((k_m / l) * Math.sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.8e-29:
		tmp = 2.0 * math.pow((k_m * ((math.sin(k_m) * math.sqrt((t_m / math.cos(k_m)))) / l)), -2.0)
	else:
		tmp = 2.0 / ((math.sin(k_m) * math.tan(k_m)) * math.pow(((k_m / l) * math.sqrt(t_m)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-29)
		tmp = Float64(2.0 * (Float64(k_m * Float64(Float64(sin(k_m) * sqrt(Float64(t_m / cos(k_m)))) / l)) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(Float64(k_m / l) * sqrt(t_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e-29)
		tmp = 2.0 * ((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) / l)) ^ -2.0);
	else
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * (((k_m / l) * sqrt(t_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.8e-29], N[(2.0 * N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.79999999999999987e-29

   1. Initial program 35.0%

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

   4. Applied egg-rr 33.8%

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

      1. associate-*r/ 33.8%

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

      2. metadata-eval 33.8%

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

      3. associate-*r* 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in k around inf 52.1%

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

      1. associate-*l/ 50.6%

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

   9. Simplified 50.6%

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

       1. pow1 50.6%

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

       2. associate-*l* 51.8%

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

   11. Applied egg-rr 51.8%

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

       1. unpow1 51.8%

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

   13. Simplified 51.8%

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

       1. div-inv 51.8%

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

       2. pow-flip 52.2%

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

       3. associate-/l* 53.9%

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

       4. metadata-eval 53.9%

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

   15. Applied egg-rr 53.9%

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

   if 1.79999999999999987e-29 < k

   1. Initial program 24.5%

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

   4. Applied egg-rr 23.3%

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

      1. associate-*r/ 23.3%

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

      2. metadata-eval 23.3%

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

      3. associate-*r* 23.3%

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

   6. Simplified 23.3%

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

      1. *-un-lft-identity 23.3%

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

      2. *-commutative 23.3%

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

      3. unpow-prod-down 23.4%

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

      4. pow2 23.4%

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

      5. add-sqr-sqrt 36.5%

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

   8. Applied egg-rr 36.5%

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

      1. *-lft-identity 36.5%

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

      2. times-frac 36.3%

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

      3. *-commutative 36.3%

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

      4. times-frac 44.9%

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

   10. Simplified 44.9%

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

   11. Taylor expanded in t around 0 53.4%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.75 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot {\left(k\_m \cdot \frac{\sin k\_m \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}{\ell}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot \left(t\_m \cdot \frac{k\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.75e-48)
    (* 2.0 (pow (* k_m (/ (* (sin k_m) (sqrt (/ t_m (cos k_m)))) l)) -2.0))
    (/ 2.0 (* (* (sin k_m) (tan k_m)) (* (/ k_m l) (* t_m (/ k_m l))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.75e-48) {
		tmp = 2.0 * pow((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) / l)), -2.0);
	} else {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.75d-48) then
        tmp = 2.0d0 * ((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) / l)) ** (-2.0d0))
    else
        tmp = 2.0d0 / ((sin(k_m) * tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.75e-48) {
		tmp = 2.0 * Math.pow((k_m * ((Math.sin(k_m) * Math.sqrt((t_m / Math.cos(k_m)))) / l)), -2.0);
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.75e-48:
		tmp = 2.0 * math.pow((k_m * ((math.sin(k_m) * math.sqrt((t_m / math.cos(k_m)))) / l)), -2.0)
	else:
		tmp = 2.0 / ((math.sin(k_m) * math.tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.75e-48)
		tmp = Float64(2.0 * (Float64(k_m * Float64(Float64(sin(k_m) * sqrt(Float64(t_m / cos(k_m)))) / l)) ^ -2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64(k_m / l) * Float64(t_m * Float64(k_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.75e-48)
		tmp = 2.0 * ((k_m * ((sin(k_m) * sqrt((t_m / cos(k_m)))) / l)) ^ -2.0);
	else
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.75e-48], N[(2.0 * N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(t$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.75000000000000023e-48

   1. Initial program 34.1%

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

   4. Applied egg-rr 34.5%

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

      1. associate-*r/ 34.5%

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

      2. metadata-eval 34.5%

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

      3. associate-*r* 34.5%

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

   6. Simplified 34.5%

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

   7. Taylor expanded in k around inf 53.3%

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

      1. associate-*l/ 51.8%

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

   9. Simplified 51.8%

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

       1. pow1 51.8%

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

       2. associate-*l* 52.9%

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

   11. Applied egg-rr 52.9%

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

       1. unpow1 52.9%

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

   13. Simplified 52.9%

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

       1. div-inv 52.9%

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

       2. pow-flip 53.3%

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

       3. associate-/l* 55.0%

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

       4. metadata-eval 55.0%

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

   15. Applied egg-rr 55.0%

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

   if 2.75000000000000023e-48 < k

   1. Initial program 27.3%

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

   4. Applied egg-rr 22.1%

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

      1. associate-*r/ 22.1%

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

      2. metadata-eval 22.1%

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

      3. associate-*r* 22.1%

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

   6. Simplified 22.1%

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

      1. *-un-lft-identity 22.1%

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

      2. *-commutative 22.1%

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

      3. unpow-prod-down 22.1%

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

      4. pow2 22.1%

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

      5. add-sqr-sqrt 34.5%

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

   8. Applied egg-rr 34.5%

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

      1. *-lft-identity 34.5%

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

      2. times-frac 34.3%

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

      3. *-commutative 34.3%

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

      4. times-frac 42.5%

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

   10. Simplified 42.5%

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

   11. Taylor expanded in t around 0 50.5%

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

       1. unpow2 50.5%

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

       2. swap-sqr 46.6%

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

       3. add-sqr-sqrt 86.3%

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

       4. associate-*l* 98.9%

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

   13. Applied egg-rr 98.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot {\left(k \cdot \frac{\sin k \cdot \sqrt{\frac{t}{\cos k}}}{\ell}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\sin k\_m \cdot \left(\tan k\_m \cdot \left(t\_m \cdot {\left(\frac{k\_m}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot \left(t\_m \cdot \frac{k\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4e-112)
    (/ 2.0 (* (sin k_m) (* (tan k_m) (* t_m (pow (/ k_m l) 2.0)))))
    (/ 2.0 (* (* (sin k_m) (tan k_m)) (* (/ k_m l) (* t_m (/ k_m l))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-112) {
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * (t_m * pow((k_m / l), 2.0))));
	} else {
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4d-112) then
        tmp = 2.0d0 / (sin(k_m) * (tan(k_m) * (t_m * ((k_m / l) ** 2.0d0))))
    else
        tmp = 2.0d0 / ((sin(k_m) * tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-112) {
		tmp = 2.0 / (Math.sin(k_m) * (Math.tan(k_m) * (t_m * Math.pow((k_m / l), 2.0))));
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 4e-112:
		tmp = 2.0 / (math.sin(k_m) * (math.tan(k_m) * (t_m * math.pow((k_m / l), 2.0))))
	else:
		tmp = 2.0 / ((math.sin(k_m) * math.tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-112)
		tmp = Float64(2.0 / Float64(sin(k_m) * Float64(tan(k_m) * Float64(t_m * (Float64(k_m / l) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64(k_m / l) * Float64(t_m * Float64(k_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 4e-112)
		tmp = 2.0 / (sin(k_m) * (tan(k_m) * (t_m * ((k_m / l) ^ 2.0))));
	else
		tmp = 2.0 / ((sin(k_m) * tan(k_m)) * ((k_m / l) * (t_m * (k_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4e-112], N[(2.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(t$95$m * N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(t$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.9999999999999998e-112

   1. Initial program 34.7%

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

   4. Applied egg-rr 34.1%

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

      1. associate-*r/ 34.1%

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

      2. metadata-eval 34.1%

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

      3. associate-*r* 34.0%

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

   6. Simplified 34.0%

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

      1. *-un-lft-identity 34.0%

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

      2. *-commutative 34.0%

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

      3. unpow-prod-down 34.0%

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

      4. pow2 34.0%

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

      5. add-sqr-sqrt 40.5%

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

   8. Applied egg-rr 40.5%

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

      1. *-lft-identity 40.5%

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

      2. times-frac 38.2%

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

      3. *-commutative 38.2%

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

      4. times-frac 45.8%

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

   10. Simplified 45.8%

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

       1. div-inv 45.8%

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

       2. associate-*l* 47.0%

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

       3. *-commutative 47.0%

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

       4. unpow-prod-down 45.9%

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

       5. pow1 45.9%

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

       6. pow-div 53.4%

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

       7. metadata-eval 53.4%

          \[\leadsto 2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot \left({\left({t}^{\color{blue}{0.5}}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)} \]

       8. pow1/2 53.4%

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

       9. pow2 53.4%

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

       10. add-sqr-sqrt 93.8%

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

   12. Applied egg-rr 93.8%

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

       1. associate-*r/ 93.8%

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

       2. metadata-eval 93.8%

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

   14. Simplified 93.8%

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

   if 3.9999999999999998e-112 < k

   1. Initial program 27.2%

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

   4. Applied egg-rr 24.8%

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

      1. associate-*r/ 24.8%

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

      2. metadata-eval 24.8%

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

      3. associate-*r* 24.8%

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

   6. Simplified 24.8%

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

      1. *-un-lft-identity 24.8%

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

      2. *-commutative 24.8%

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

      3. unpow-prod-down 24.8%

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

      4. pow2 24.8%

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

      5. add-sqr-sqrt 35.5%

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

   8. Applied egg-rr 35.5%

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

      1. *-lft-identity 35.5%

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

      2. times-frac 34.7%

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

      3. *-commutative 34.7%

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

      4. times-frac 42.4%

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

   10. Simplified 42.4%

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

   11. Taylor expanded in t around 0 49.2%

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

       1. unpow2 49.2%

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

       2. swap-sqr 44.8%

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

       3. add-sqr-sqrt 85.7%

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

       4. associate-*l* 97.9%

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

   13. Applied egg-rr 97.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * ((k_m / l) * (t_m * (k_m / l)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

4. Applied egg-rr 31.0%

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

   1. associate-*r/ 31.0%

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

   2. metadata-eval 31.0%

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

   3. associate-*r* 31.0%

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

6. Simplified 31.0%

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

   1. *-un-lft-identity 31.0%

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

   2. *-commutative 31.0%

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

   3. unpow-prod-down 31.0%

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

   4. pow2 31.0%

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

   5. add-sqr-sqrt 38.8%

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

8. Applied egg-rr 38.8%

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

   1. *-lft-identity 38.8%

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

   2. times-frac 37.0%

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

   3. *-commutative 37.0%

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

   4. times-frac 44.7%

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

10. Simplified 44.7%

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

11. Taylor expanded in t around 0 52.6%

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

    1. unpow2 52.6%

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

    2. swap-sqr 49.8%

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

    3. add-sqr-sqrt 88.4%

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

    4. associate-*l* 95.3%

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

13. Applied egg-rr 95.3%

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

14. Final simplification 95.3%

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

Alternative 6: 72.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((t_m * Math.pow((k_m / l), 2.0)) * (k_m * Math.sin(k_m))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

4. Applied egg-rr 31.0%

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

   1. associate-*r/ 31.0%

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

   2. metadata-eval 31.0%

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

   3. associate-*r* 31.0%

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

6. Simplified 31.0%

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

   1. *-un-lft-identity 31.0%

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

   2. *-commutative 31.0%

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

   3. unpow-prod-down 31.0%

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

   4. pow2 31.0%

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

   5. add-sqr-sqrt 38.8%

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

8. Applied egg-rr 38.8%

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

   1. *-lft-identity 38.8%

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

   2. times-frac 37.0%

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

   3. *-commutative 37.0%

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

   4. times-frac 44.7%

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

10. Simplified 44.7%

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

    1. pow1 44.7%

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

    2. associate-*l* 45.5%

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

    3. *-commutative 45.5%

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

    4. unpow-prod-down 44.0%

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

    5. pow1 44.0%

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

    6. pow-div 50.5%

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

    7. metadata-eval 50.5%

       \[\leadsto \frac{2}{{\left(\sin k \cdot \left(\tan k \cdot \left({\left({t}^{\color{blue}{0.5}}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)\right)}^{1}} \]

    8. pow1/2 50.5%

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

    9. pow2 50.5%

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

    10. add-sqr-sqrt 91.1%

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

12. Applied egg-rr 91.1%

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

    1. unpow1 91.1%

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

    2. associate-*r* 88.4%

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

14. Simplified 88.4%

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

15. Taylor expanded in k around 0 68.9%

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

16. Final simplification 68.9%

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

Alternative 7: 72.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((t_m * Math.pow((k_m / l), 2.0)) * (k_m * Math.tan(k_m))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

4. Applied egg-rr 31.0%

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

   1. associate-*r/ 31.0%

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

   2. metadata-eval 31.0%

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

   3. associate-*r* 31.0%

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

6. Simplified 31.0%

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

   1. *-un-lft-identity 31.0%

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

   2. *-commutative 31.0%

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

   3. unpow-prod-down 31.0%

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

   4. pow2 31.0%

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

   5. add-sqr-sqrt 38.8%

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

8. Applied egg-rr 38.8%

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

   1. *-lft-identity 38.8%

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

   2. times-frac 37.0%

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

   3. *-commutative 37.0%

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

   4. times-frac 44.7%

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

10. Simplified 44.7%

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

    1. pow1 44.7%

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

    2. associate-*l* 45.5%

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

    3. *-commutative 45.5%

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

    4. unpow-prod-down 44.0%

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

    5. pow1 44.0%

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

    6. pow-div 50.5%

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

    7. metadata-eval 50.5%

       \[\leadsto \frac{2}{{\left(\sin k \cdot \left(\tan k \cdot \left({\left({t}^{\color{blue}{0.5}}\right)}^{2} \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)\right)\right)}^{1}} \]

    8. pow1/2 50.5%

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

    9. pow2 50.5%

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

    10. add-sqr-sqrt 91.1%

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

12. Applied egg-rr 91.1%

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

    1. unpow1 91.1%

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

    2. associate-*r* 88.4%

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

14. Simplified 88.4%

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

15. Taylor expanded in k around 0 68.4%

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

16. Final simplification 68.4%

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

Alternative 8: 72.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((t_m * Math.pow((k_m / l), 2.0)) * Math.pow(k_m, 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

4. Applied egg-rr 31.0%

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

   1. associate-*r/ 31.0%

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

   2. metadata-eval 31.0%

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

   3. associate-*r* 31.0%

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

6. Simplified 31.0%

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

7. Taylor expanded in k around inf 67.1%

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

   1. associate-/l* 69.5%

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

9. Simplified 69.5%

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

10. Taylor expanded in k around 0 60.1%

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

    1. *-commutative 60.1%

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

    2. associate-/l* 59.7%

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

    3. unpow2 59.7%

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

    4. unpow2 59.7%

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

    5. times-frac 68.3%

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

    6. unpow2 68.3%

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

12. Simplified 68.3%

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

13. Final simplification 68.3%

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

Alternative 9: 63.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((2.0 / (t_m * Math.pow(k_m, 4.0))) * (l * l));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

3. Simplified 40.4%

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

5. Taylor expanded in k around 0 55.2%

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

6. Final simplification 55.2%

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

Alternative 10: 29.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t\_m \cdot {k\_m}^{2}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ -0.3333333333333333 (* t_m (pow k_m 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.3333333333333333 / (t_m * pow(k_m, 2.0))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((-0.3333333333333333d0) / (t_m * (k_m ** 2.0d0))))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.3333333333333333 / (t_m * Math.pow(k_m, 2.0))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (-0.3333333333333333 / (t_m * math.pow(k_m, 2.0))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(t_m * (k_m ^ 2.0)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (-0.3333333333333333 / (t_m * (k_m ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 32.2%

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

3. Simplified 40.4%

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

5. Taylor expanded in k around 0 40.4%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 28.2%

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

7. Final simplification 28.2%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot {k}^{2}} \]
8. Add Preprocessing

Alternative 11: 20.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{{\ell}^{2} \cdot -0.11666666666666667}{t\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ (* (pow l 2.0) -0.11666666666666667) t_m)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((pow(l, 2.0) * -0.11666666666666667) / t_m);
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (((l ** 2.0d0) * (-0.11666666666666667d0)) / t_m)
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((Math.pow(l, 2.0) * -0.11666666666666667) / t_m);
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((math.pow(l, 2.0) * -0.11666666666666667) / t_m)

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64((l ^ 2.0) * -0.11666666666666667) / t_m))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (((l ^ 2.0) * -0.11666666666666667) / t_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(N[Power[l, 2.0], $MachinePrecision] * -0.11666666666666667), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Simplified 40.4%

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

5. Taylor expanded in k around 0 37.5%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 21.8%

   \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative 21.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.11666666666666667} \]

   2. associate-*l/ 21.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot -0.11666666666666667}{t}} \]

8. Simplified 21.8%

   \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot -0.11666666666666667}{t}} \]
9. Add Preprocessing

Alternative 12: 20.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ -0.11666666666666667 t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((-0.11666666666666667d0) / t_m))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (-0.11666666666666667 / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.11666666666666667 / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (-0.11666666666666667 / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

3. Simplified 40.4%

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

5. Taylor expanded in k around 0 37.5%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 21.8%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 21.8%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(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))))