Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 95.5%

Time: 15.0s

Alternatives: 17

Speedup: 3.8×

• 40.1% 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: 35.4% 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: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 37000:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \frac{\sqrt{t\_m}}{\sqrt{\cos k\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{2}{t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 37000.0)
    (/
     2.0
     (pow (* (* k_m (/ (sin k_m) l)) (/ (sqrt t_m) (sqrt (cos k_m)))) 2.0))
    (*
     (pow (/ l k_m) 2.0)
     (/ 2.0 (* t_m (/ (pow (sin k_m) 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 37000.0) {
		tmp = 2.0 / pow(((k_m * (sin(k_m) / l)) * (sqrt(t_m) / sqrt(cos(k_m)))), 2.0);
	} else {
		tmp = pow((l / k_m), 2.0) * (2.0 / (t_m * (pow(sin(k_m), 2.0) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 37000.0d0) then
        tmp = 2.0d0 / (((k_m * (sin(k_m) / l)) * (sqrt(t_m) / sqrt(cos(k_m)))) ** 2.0d0)
    else
        tmp = ((l / k_m) ** 2.0d0) * (2.0d0 / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 37000.0) {
		tmp = 2.0 / Math.pow(((k_m * (Math.sin(k_m) / l)) * (Math.sqrt(t_m) / Math.sqrt(Math.cos(k_m)))), 2.0);
	} else {
		tmp = Math.pow((l / k_m), 2.0) * (2.0 / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 37000.0:
		tmp = 2.0 / math.pow(((k_m * (math.sin(k_m) / l)) * (math.sqrt(t_m) / math.sqrt(math.cos(k_m)))), 2.0)
	else:
		tmp = math.pow((l / k_m), 2.0) * (2.0 / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 37000.0)
		tmp = Float64(2.0 / (Float64(Float64(k_m * Float64(sin(k_m) / l)) * Float64(sqrt(t_m) / sqrt(cos(k_m)))) ^ 2.0));
	else
		tmp = Float64((Float64(l / k_m) ^ 2.0) * Float64(2.0 / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 37000.0)
		tmp = 2.0 / (((k_m * (sin(k_m) / l)) * (sqrt(t_m) / sqrt(cos(k_m)))) ^ 2.0);
	else
		tmp = ((l / k_m) ^ 2.0) * (2.0 / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 37000.0], N[(2.0 / N[Power[N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 37000

   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

   3. Taylor expanded in t around 0 74.1%

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

      1. associate-/l* 75.9%

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

   5. Simplified 75.9%

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

      1. div-inv 75.9%

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

      2. add-sqr-sqrt 35.9%

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

      3. pow2 35.9%

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

   7. Applied egg-rr 42.8%

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

      1. associate-*r/ 42.8%

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

      2. metadata-eval 42.8%

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

      3. times-frac 42.8%

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

      4. associate-*r* 43.3%

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

   9. Simplified 43.3%

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

   if 37000 < k

   1. Initial program 30.5%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in t around 0 75.6%

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

      1. associate-/l* 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*l/ 75.5%

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

      2. pow2 75.5%

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

      3. associate-/l* 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. *-commutative 75.5%

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

       2. times-frac 75.8%

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

       3. unpow2 75.8%

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

       4. unpow2 75.8%

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

       5. times-frac 93.1%

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

       6. unpow2 93.1%

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

   11. Simplified 93.1%

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

Alternative 2: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 37000:\\ \;\;\;\;\frac{2}{{\left(k\_m \cdot \frac{\sin k\_m \cdot \sqrt{t\_m}}{\ell \cdot \sqrt{\cos k\_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{2}{t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 37000.0)
    (/
     2.0
     (pow (* k_m (/ (* (sin k_m) (sqrt t_m)) (* l (sqrt (cos k_m))))) 2.0))
    (*
     (pow (/ l k_m) 2.0)
     (/ 2.0 (* t_m (/ (pow (sin k_m) 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 37000.0) {
		tmp = 2.0 / pow((k_m * ((sin(k_m) * sqrt(t_m)) / (l * sqrt(cos(k_m))))), 2.0);
	} else {
		tmp = pow((l / k_m), 2.0) * (2.0 / (t_m * (pow(sin(k_m), 2.0) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 37000.0d0) then
        tmp = 2.0d0 / ((k_m * ((sin(k_m) * sqrt(t_m)) / (l * sqrt(cos(k_m))))) ** 2.0d0)
    else
        tmp = ((l / k_m) ** 2.0d0) * (2.0d0 / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 37000.0) {
		tmp = 2.0 / Math.pow((k_m * ((Math.sin(k_m) * Math.sqrt(t_m)) / (l * Math.sqrt(Math.cos(k_m))))), 2.0);
	} else {
		tmp = Math.pow((l / k_m), 2.0) * (2.0 / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 37000.0:
		tmp = 2.0 / math.pow((k_m * ((math.sin(k_m) * math.sqrt(t_m)) / (l * math.sqrt(math.cos(k_m))))), 2.0)
	else:
		tmp = math.pow((l / k_m), 2.0) * (2.0 / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 37000.0)
		tmp = Float64(2.0 / (Float64(k_m * Float64(Float64(sin(k_m) * sqrt(t_m)) / Float64(l * sqrt(cos(k_m))))) ^ 2.0));
	else
		tmp = Float64((Float64(l / k_m) ^ 2.0) * Float64(2.0 / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 37000.0)
		tmp = 2.0 / ((k_m * ((sin(k_m) * sqrt(t_m)) / (l * sqrt(cos(k_m))))) ^ 2.0);
	else
		tmp = ((l / k_m) ^ 2.0) * (2.0 / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 37000.0], N[(2.0 / N[Power[N[(k$95$m * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 37000

   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

   3. Taylor expanded in t around 0 74.1%

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

      1. associate-/l* 75.9%

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

   5. Simplified 75.9%

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

      1. add-sqr-sqrt 35.9%

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

      2. pow2 35.9%

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

   7. Applied egg-rr 42.8%

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

   if 37000 < k

   1. Initial program 30.5%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in t around 0 75.6%

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

      1. associate-/l* 75.5%

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

   7. Simplified 75.5%

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

      1. associate-*l/ 75.5%

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

      2. pow2 75.5%

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

      3. associate-/l* 75.5%

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

   9. Applied egg-rr 75.5%

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

       1. *-commutative 75.5%

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

       2. times-frac 75.8%

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

       3. unpow2 75.8%

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

       4. unpow2 75.8%

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

       5. times-frac 93.1%

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

       6. unpow2 93.1%

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

   11. Simplified 93.1%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k\_m}{t\_m}} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{2}{t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.2e-22)
    (pow (* (sqrt (/ (cos k_m) t_m)) (* (/ l k_m) (/ (sqrt 2.0) k_m))) 2.0)
    (*
     (pow (/ l k_m) 2.0)
     (/ 2.0 (* t_m (/ (pow (sin k_m) 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-22) {
		tmp = pow((sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = pow((l / k_m), 2.0) * (2.0 / (t_m * (pow(sin(k_m), 2.0) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.2d-22) then
        tmp = (sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0d0) / k_m))) ** 2.0d0
    else
        tmp = ((l / k_m) ** 2.0d0) * (2.0d0 / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-22) {
		tmp = Math.pow((Math.sqrt((Math.cos(k_m) / t_m)) * ((l / k_m) * (Math.sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = Math.pow((l / k_m), 2.0) * (2.0 / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.2e-22:
		tmp = math.pow((math.sqrt((math.cos(k_m) / t_m)) * ((l / k_m) * (math.sqrt(2.0) / k_m))), 2.0)
	else:
		tmp = math.pow((l / k_m), 2.0) * (2.0 / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.2e-22)
		tmp = Float64(sqrt(Float64(cos(k_m) / t_m)) * Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = Float64((Float64(l / k_m) ^ 2.0) * Float64(2.0 / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.2e-22)
		tmp = (sqrt((cos(k_m) / t_m)) * ((l / k_m) * (sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = ((l / k_m) ^ 2.0) * (2.0 / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.2e-22], N[Power[N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000001e-22

   1. Initial program 32.3%

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in k around inf 47.4%

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

      1. times-frac 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in k around 0 41.2%

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

   if 1.20000000000000001e-22 < k

   1. Initial program 30.4%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in t around 0 77.8%

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

      1. associate-/l* 77.7%

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

   7. Simplified 77.7%

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

      1. associate-*l/ 77.8%

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

      2. pow2 77.8%

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

      3. associate-/l* 77.8%

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

   9. Applied egg-rr 77.8%

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

       1. *-commutative 77.8%

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

       2. times-frac 78.0%

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

       3. unpow2 78.0%

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

       4. unpow2 78.0%

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

       5. times-frac 92.6%

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

       6. unpow2 92.6%

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

   11. Simplified 92.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-22}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{2}{t \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-22}:\\ \;\;\;\;{\left(\sqrt{t\_2} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{t\_2}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 1.15e-22)
      (pow (* (sqrt t_2) (* (/ l k_m) (/ (sqrt 2.0) k_m))) 2.0)
      (* 2.0 (* (pow (/ l k_m) 2.0) (/ t_2 (pow (sin 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) {
	double t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 1.15e-22) {
		tmp = pow((sqrt(t_2) * ((l / k_m) * (sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * (t_2 / pow(sin(k_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) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 1.15d-22) then
        tmp = (sqrt(t_2) * ((l / k_m) * (sqrt(2.0d0) / k_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * (t_2 / (sin(k_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 t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 1.15e-22) {
		tmp = Math.pow((Math.sqrt(t_2) * ((l / k_m) * (Math.sqrt(2.0) / k_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * (t_2 / Math.pow(Math.sin(k_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):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 1.15e-22:
		tmp = math.pow((math.sqrt(t_2) * ((l / k_m) * (math.sqrt(2.0) / k_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * (t_2 / math.pow(math.sin(k_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)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 1.15e-22)
		tmp = Float64(sqrt(t_2) * Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(t_2 / (sin(k_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)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 1.15e-22)
		tmp = (sqrt(t_2) * ((l / k_m) * (sqrt(2.0) / k_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (t_2 / (sin(k_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_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.15e-22], N[Power[N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$2 / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-22

   1. Initial program 32.3%

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in k around inf 47.4%

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

      1. times-frac 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in k around 0 41.2%

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

   if 1.1499999999999999e-22 < k

   1. Initial program 30.4%

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

   4. Applied egg-rr 24.5%

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

   5. Taylor expanded in k around inf 55.7%

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

      1. times-frac 55.8%

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

   7. Simplified 55.8%

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

   8. Taylor expanded in l around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. unpow2 77.6%

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

      4. rem-square-sqrt 77.8%

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

      5. associate-*r* 77.8%

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

      6. *-commutative 77.8%

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

      7. associate-*r/ 77.8%

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

      8. *-commutative 77.8%

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

      9. *-commutative 77.8%

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

      10. times-frac 77.9%

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

      11. associate-/r* 77.9%

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

      12. unpow2 77.9%

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

      13. unpow2 77.9%

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

      14. times-frac 92.6%

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

      15. unpow2 92.6%

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

   10. Simplified 92.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-22}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k\_m}\\ t_3 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt{t\_3} \cdot \left(\frac{\ell}{k\_m} \cdot t\_2\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot {\left(t\_2 \cdot \frac{\ell}{\sin k\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_2 (/ (sqrt 2.0) k_m)) (t_3 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 2.45e-6)
      (pow (* (sqrt t_3) (* (/ l k_m) t_2)) 2.0)
      (* t_3 (pow (* t_2 (/ l (sin 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) {
	double t_2 = sqrt(2.0) / k_m;
	double t_3 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 2.45e-6) {
		tmp = pow((sqrt(t_3) * ((l / k_m) * t_2)), 2.0);
	} else {
		tmp = t_3 * pow((t_2 * (l / sin(k_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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sqrt(2.0d0) / k_m
    t_3 = cos(k_m) / t_m
    if (k_m <= 2.45d-6) then
        tmp = (sqrt(t_3) * ((l / k_m) * t_2)) ** 2.0d0
    else
        tmp = t_3 * ((t_2 * (l / sin(k_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 t_2 = Math.sqrt(2.0) / k_m;
	double t_3 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 2.45e-6) {
		tmp = Math.pow((Math.sqrt(t_3) * ((l / k_m) * t_2)), 2.0);
	} else {
		tmp = t_3 * Math.pow((t_2 * (l / Math.sin(k_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):
	t_2 = math.sqrt(2.0) / k_m
	t_3 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 2.45e-6:
		tmp = math.pow((math.sqrt(t_3) * ((l / k_m) * t_2)), 2.0)
	else:
		tmp = t_3 * math.pow((t_2 * (l / math.sin(k_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)
	t_2 = Float64(sqrt(2.0) / k_m)
	t_3 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 2.45e-6)
		tmp = Float64(sqrt(t_3) * Float64(Float64(l / k_m) * t_2)) ^ 2.0;
	else
		tmp = Float64(t_3 * (Float64(t_2 * Float64(l / sin(k_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)
	t_2 = sqrt(2.0) / k_m;
	t_3 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 2.45e-6)
		tmp = (sqrt(t_3) * ((l / k_m) * t_2)) ^ 2.0;
	else
		tmp = t_3 * ((t_2 * (l / sin(k_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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.45e-6], N[Power[N[(N[Sqrt[t$95$3], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(t$95$3 * N[Power[N[(t$95$2 * N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.44999999999999984e-6

   1. Initial program 32.8%

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

   4. Applied egg-rr 31.8%

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

   5. Taylor expanded in k around inf 48.0%

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

      1. times-frac 50.7%

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

   7. Simplified 50.7%

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

   8. Taylor expanded in k around 0 42.0%

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

   if 2.44999999999999984e-6 < k

   1. Initial program 28.5%

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

   4. Applied egg-rr 21.9%

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

   5. Taylor expanded in k around inf 54.7%

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

      1. times-frac 54.8%

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

   7. Simplified 54.8%

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

      1. *-commutative 54.8%

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

      2. unpow-prod-down 48.5%

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

      3. pow2 48.5%

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

      4. add-sqr-sqrt 91.8%

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

      5. frac-times 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. *-commutative 91.7%

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

       2. times-frac 91.7%

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

   11. Simplified 91.7%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\sqrt{2}}{k} \cdot \frac{\ell}{\sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sin k\_m \cdot \left(k\_m \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\frac{\sqrt{2}}{k\_m} \cdot \frac{\ell}{\sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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-12)
    (pow (* l (/ (sqrt 2.0) (* (sin k_m) (* k_m (sqrt t_m))))) 2.0)
    (* (/ (cos k_m) t_m) (pow (* (/ (sqrt 2.0) k_m) (/ l (sin 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) {
	double tmp;
	if (k_m <= 1.8e-12) {
		tmp = pow((l * (sqrt(2.0) / (sin(k_m) * (k_m * sqrt(t_m))))), 2.0);
	} else {
		tmp = (cos(k_m) / t_m) * pow(((sqrt(2.0) / k_m) * (l / sin(k_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-12) then
        tmp = (l * (sqrt(2.0d0) / (sin(k_m) * (k_m * sqrt(t_m))))) ** 2.0d0
    else
        tmp = (cos(k_m) / t_m) * (((sqrt(2.0d0) / k_m) * (l / sin(k_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-12) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (Math.sin(k_m) * (k_m * Math.sqrt(t_m))))), 2.0);
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow(((Math.sqrt(2.0) / k_m) * (l / Math.sin(k_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-12:
		tmp = math.pow((l * (math.sqrt(2.0) / (math.sin(k_m) * (k_m * math.sqrt(t_m))))), 2.0)
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow(((math.sqrt(2.0) / k_m) * (l / math.sin(k_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-12)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(sin(k_m) * Float64(k_m * sqrt(t_m))))) ^ 2.0;
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(Float64(sqrt(2.0) / k_m) * Float64(l / sin(k_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-12)
		tmp = (l * (sqrt(2.0) / (sin(k_m) * (k_m * sqrt(t_m))))) ^ 2.0;
	else
		tmp = (cos(k_m) / t_m) * (((sqrt(2.0) / k_m) * (l / sin(k_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-12], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-12

   1. Initial program 32.8%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. associate-*r* 73.7%

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

      3. times-frac 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in k around 0 68.0%

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

      1. pow1 68.0%

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

   10. Applied egg-rr 43.7%

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

       1. unpow1 43.7%

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

       2. associate-/l/ 43.7%

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

   12. Simplified 43.7%

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

   if 1.8e-12 < k

   1. Initial program 28.7%

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

   4. Applied egg-rr 22.5%

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

   5. Taylor expanded in k around inf 55.2%

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

      1. times-frac 55.3%

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

   7. Simplified 55.3%

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

      1. *-commutative 55.3%

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

      2. unpow-prod-down 49.4%

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

      3. pow2 49.4%

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

      4. add-sqr-sqrt 92.1%

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

      5. frac-times 92.0%

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

   9. Applied egg-rr 92.0%

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

       1. *-commutative 92.0%

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

       2. times-frac 92.1%

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

   11. Simplified 92.1%

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

Alternative 7: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sin k\_m \cdot \left(k\_m \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k\_m \cdot \frac{{\left(\frac{\sqrt{2}}{k\_m} \cdot \frac{\ell}{\sin k\_m}\right)}^{2}}{t\_m}\\ \end{array} \end{array} \]

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-12)
    (pow (* l (/ (sqrt 2.0) (* (sin k_m) (* k_m (sqrt t_m))))) 2.0)
    (* (cos k_m) (/ (pow (* (/ (sqrt 2.0) k_m) (/ l (sin k_m))) 2.0) t_m)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-12) {
		tmp = pow((l * (sqrt(2.0) / (sin(k_m) * (k_m * sqrt(t_m))))), 2.0);
	} else {
		tmp = cos(k_m) * (pow(((sqrt(2.0) / k_m) * (l / sin(k_m))), 2.0) / t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d-12) then
        tmp = (l * (sqrt(2.0d0) / (sin(k_m) * (k_m * sqrt(t_m))))) ** 2.0d0
    else
        tmp = cos(k_m) * ((((sqrt(2.0d0) / k_m) * (l / sin(k_m))) ** 2.0d0) / t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-12) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (Math.sin(k_m) * (k_m * Math.sqrt(t_m))))), 2.0);
	} else {
		tmp = Math.cos(k_m) * (Math.pow(((Math.sqrt(2.0) / k_m) * (l / Math.sin(k_m))), 2.0) / t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.8e-12:
		tmp = math.pow((l * (math.sqrt(2.0) / (math.sin(k_m) * (k_m * math.sqrt(t_m))))), 2.0)
	else:
		tmp = math.cos(k_m) * (math.pow(((math.sqrt(2.0) / k_m) * (l / math.sin(k_m))), 2.0) / t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-12)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(sin(k_m) * Float64(k_m * sqrt(t_m))))) ^ 2.0;
	else
		tmp = Float64(cos(k_m) * Float64((Float64(Float64(sqrt(2.0) / k_m) * Float64(l / sin(k_m))) ^ 2.0) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e-12)
		tmp = (l * (sqrt(2.0) / (sin(k_m) * (k_m * sqrt(t_m))))) ^ 2.0;
	else
		tmp = cos(k_m) * ((((sqrt(2.0) / k_m) * (l / sin(k_m))) ^ 2.0) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.8e-12], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-12

   1. Initial program 32.8%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in t around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. associate-*r* 73.7%

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

      3. times-frac 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in k around 0 68.0%

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

      1. pow1 68.0%

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

   10. Applied egg-rr 43.7%

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

       1. unpow1 43.7%

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

       2. associate-/l/ 43.7%

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

   12. Simplified 43.7%

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

   if 1.8e-12 < k

   1. Initial program 28.7%

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

   4. Applied egg-rr 22.5%

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

   5. Taylor expanded in k around inf 55.2%

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

      1. times-frac 55.3%

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

   7. Simplified 55.3%

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

      1. *-commutative 55.3%

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

      2. unpow-prod-down 49.4%

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

      3. pow2 49.4%

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

      4. add-sqr-sqrt 92.1%

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

      5. frac-times 92.0%

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

   9. Applied egg-rr 92.0%

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

       1. *-commutative 92.0%

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

       2. times-frac 92.1%

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

   11. Simplified 92.1%

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

       1. associate-*l/ 92.1%

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

       2. frac-times 92.1%

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

   13. Applied egg-rr 92.1%

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

       1. associate-/l* 92.0%

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

       2. times-frac 92.1%

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

   15. Simplified 92.1%

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

Alternative 8: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-46}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2}}{\sin k\_m \cdot \left(k\_m \cdot \sqrt{t\_m}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k\_m}{k\_m \cdot k\_m}}{t\_m \cdot {\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

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 1.7e-46)
    (pow (* l (/ (sqrt 2.0) (* (sin k_m) (* k_m (sqrt t_m))))) 2.0)
    (*
     (* 2.0 (/ (/ (cos k_m) (* k_m k_m)) (* t_m (pow (sin k_m) 2.0))))
     (* 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) {
	double tmp;
	if (k_m <= 1.7e-46) {
		tmp = pow((l * (sqrt(2.0) / (sin(k_m) * (k_m * sqrt(t_m))))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / (k_m * k_m)) / (t_m * pow(sin(k_m), 2.0)))) * (l * 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 <= 1.7d-46) then
        tmp = (l * (sqrt(2.0d0) / (sin(k_m) * (k_m * sqrt(t_m))))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) / (k_m * k_m)) / (t_m * (sin(k_m) ** 2.0d0)))) * (l * 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 <= 1.7e-46) {
		tmp = Math.pow((l * (Math.sqrt(2.0) / (Math.sin(k_m) * (k_m * Math.sqrt(t_m))))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / (k_m * k_m)) / (t_m * Math.pow(Math.sin(k_m), 2.0)))) * (l * 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 <= 1.7e-46:
		tmp = math.pow((l * (math.sqrt(2.0) / (math.sin(k_m) * (k_m * math.sqrt(t_m))))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / (k_m * k_m)) / (t_m * math.pow(math.sin(k_m), 2.0)))) * (l * 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 <= 1.7e-46)
		tmp = Float64(l * Float64(sqrt(2.0) / Float64(sin(k_m) * Float64(k_m * sqrt(t_m))))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) / Float64(k_m * k_m)) / Float64(t_m * (sin(k_m) ^ 2.0)))) * Float64(l * 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 <= 1.7e-46)
		tmp = (l * (sqrt(2.0) / (sin(k_m) * (k_m * sqrt(t_m))))) ^ 2.0;
	else
		tmp = (2.0 * ((cos(k_m) / (k_m * k_m)) / (t_m * (sin(k_m) ^ 2.0)))) * (l * 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, 1.7e-46], N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.69999999999999998e-46

   1. Initial program 33.1%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. associate-*r* 72.6%

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

      3. times-frac 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in k around 0 66.7%

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

      1. pow1 66.7%

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

   10. Applied egg-rr 43.5%

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

       1. unpow1 43.5%

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

       2. associate-/l/ 43.5%

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

   12. Simplified 43.5%

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

   if 1.69999999999999998e-46 < k

   1. Initial program 28.5%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in t around 0 79.2%

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

      1. associate-/r* 79.2%

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

   7. Simplified 79.2%

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

      1. unpow2 79.2%

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

   9. Applied egg-rr 79.2%

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

Alternative 9: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k\_m}{k\_m \cdot k\_m}}{t\_m \cdot {\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

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 (<= (* l l) 1.2e-135)
    (* (/ (cos k_m) t_m) (pow (* (/ l k_m) (/ (sqrt 2.0) k_m)) 2.0))
    (*
     (* 2.0 (/ (/ (cos k_m) (* k_m k_m)) (* t_m (pow (sin k_m) 2.0))))
     (* 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) {
	double tmp;
	if ((l * l) <= 1.2e-135) {
		tmp = (cos(k_m) / t_m) * pow(((l / k_m) * (sqrt(2.0) / k_m)), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) / (k_m * k_m)) / (t_m * pow(sin(k_m), 2.0)))) * (l * 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 ((l * l) <= 1.2d-135) then
        tmp = (cos(k_m) / t_m) * (((l / k_m) * (sqrt(2.0d0) / k_m)) ** 2.0d0)
    else
        tmp = (2.0d0 * ((cos(k_m) / (k_m * k_m)) / (t_m * (sin(k_m) ** 2.0d0)))) * (l * 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 ((l * l) <= 1.2e-135) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow(((l / k_m) * (Math.sqrt(2.0) / k_m)), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) / (k_m * k_m)) / (t_m * Math.pow(Math.sin(k_m), 2.0)))) * (l * 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 (l * l) <= 1.2e-135:
		tmp = (math.cos(k_m) / t_m) * math.pow(((l / k_m) * (math.sqrt(2.0) / k_m)), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) / (k_m * k_m)) / (t_m * math.pow(math.sin(k_m), 2.0)))) * (l * 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 (Float64(l * l) <= 1.2e-135)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) / Float64(k_m * k_m)) / Float64(t_m * (sin(k_m) ^ 2.0)))) * Float64(l * 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 ((l * l) <= 1.2e-135)
		tmp = (cos(k_m) / t_m) * (((l / k_m) * (sqrt(2.0) / k_m)) ^ 2.0);
	else
		tmp = (2.0 * ((cos(k_m) / (k_m * k_m)) / (t_m * (sin(k_m) ^ 2.0)))) * (l * 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[N[(l * l), $MachinePrecision], 1.2e-135], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.1999999999999999e-135

   1. Initial program 21.6%

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

   4. Applied egg-rr 24.6%

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

   5. Taylor expanded in k around inf 44.9%

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

      1. times-frac 50.4%

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

   7. Simplified 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow-prod-down 46.6%

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

      3. pow2 46.6%

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

      4. add-sqr-sqrt 91.3%

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

      5. frac-times 88.3%

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

   9. Applied egg-rr 88.3%

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

       1. *-commutative 88.3%

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

       2. times-frac 91.3%

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

   11. Simplified 91.3%

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

   12. Taylor expanded in k around 0 91.0%

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

   if 1.1999999999999999e-135 < (*.f64 l l)

   1. Initial program 38.3%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in t around 0 77.8%

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

      1. associate-/r* 78.1%

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

   7. Simplified 78.1%

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

      1. unpow2 78.1%

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

   9. Applied egg-rr 78.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{k \cdot k}}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{k\_m} \cdot \frac{\ell}{\sin k\_m}\right)}^{2} \cdot \frac{1}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {k\_m}^{2}}{\cos k\_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

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 (<= (* l l) 5e-315)
    (* (pow (* (/ (sqrt 2.0) k_m) (/ l (sin k_m))) 2.0) (/ 1.0 t_m))
    (/
     2.0
     (* (* k_m k_m) (/ (* t_m (pow k_m 2.0)) (* (cos k_m) (pow l 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 ((l * l) <= 5e-315) {
		tmp = pow(((sqrt(2.0) / k_m) * (l / sin(k_m))), 2.0) * (1.0 / t_m);
	} else {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * pow(k_m, 2.0)) / (cos(k_m) * pow(l, 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 ((l * l) <= 5d-315) then
        tmp = (((sqrt(2.0d0) / k_m) * (l / sin(k_m))) ** 2.0d0) * (1.0d0 / t_m)
    else
        tmp = 2.0d0 / ((k_m * k_m) * ((t_m * (k_m ** 2.0d0)) / (cos(k_m) * (l ** 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 ((l * l) <= 5e-315) {
		tmp = Math.pow(((Math.sqrt(2.0) / k_m) * (l / Math.sin(k_m))), 2.0) * (1.0 / t_m);
	} else {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * Math.pow(k_m, 2.0)) / (Math.cos(k_m) * Math.pow(l, 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 (l * l) <= 5e-315:
		tmp = math.pow(((math.sqrt(2.0) / k_m) * (l / math.sin(k_m))), 2.0) * (1.0 / t_m)
	else:
		tmp = 2.0 / ((k_m * k_m) * ((t_m * math.pow(k_m, 2.0)) / (math.cos(k_m) * math.pow(l, 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 (Float64(l * l) <= 5e-315)
		tmp = Float64((Float64(Float64(sqrt(2.0) / k_m) * Float64(l / sin(k_m))) ^ 2.0) * Float64(1.0 / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (k_m ^ 2.0)) / Float64(cos(k_m) * (l ^ 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 ((l * l) <= 5e-315)
		tmp = (((sqrt(2.0) / k_m) * (l / sin(k_m))) ^ 2.0) * (1.0 / t_m);
	else
		tmp = 2.0 / ((k_m * k_m) * ((t_m * (k_m ^ 2.0)) / (cos(k_m) * (l ^ 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[N[(l * l), $MachinePrecision], 5e-315], N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000023e-315

   1. Initial program 21.6%

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

   4. Applied egg-rr 23.4%

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

   5. Taylor expanded in k around inf 43.6%

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

      1. times-frac 51.3%

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

   7. Simplified 51.3%

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

      1. *-commutative 51.3%

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

      2. unpow-prod-down 47.3%

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

      3. pow2 47.3%

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

      4. add-sqr-sqrt 91.8%

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

      5. frac-times 87.6%

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

   9. Applied egg-rr 87.6%

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

       1. *-commutative 87.6%

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

       2. times-frac 91.8%

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

   11. Simplified 91.8%

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

   12. Taylor expanded in k around 0 91.8%

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

   if 5.0000000023e-315 < (*.f64 l l)

   1. Initial program 35.6%

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

   3. Taylor expanded in t around 0 79.2%

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

      1. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

      1. unpow2 79.5%

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

   7. Applied egg-rr 79.4%

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

   8. Taylor expanded in k around 0 68.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{k} \cdot \frac{\ell}{\sin k}\right)}^{2} \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{t\_m}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m \cdot {\sin k\_m}^{2}}\\ \end{array} \end{array} \]

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 8.5e+27)
    (pow (* l (/ (sqrt (/ 2.0 t_m)) (pow k_m 2.0))) 2.0)
    (* 2.0 (/ (pow (/ l k_m) 2.0) (* t_m (pow (sin 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) {
	double tmp;
	if (k_m <= 8.5e+27) {
		tmp = pow((l * (sqrt((2.0 / t_m)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) / (t_m * pow(sin(k_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 <= 8.5d+27) then
        tmp = (l * (sqrt((2.0d0 / t_m)) / (k_m ** 2.0d0))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) / (t_m * (sin(k_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 <= 8.5e+27) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / t_m)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) / (t_m * Math.pow(Math.sin(k_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 <= 8.5e+27:
		tmp = math.pow((l * (math.sqrt((2.0 / t_m)) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) / (t_m * math.pow(math.sin(k_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 <= 8.5e+27)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / t_m)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) / Float64(t_m * (sin(k_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 <= 8.5e+27)
		tmp = (l * (sqrt((2.0 / t_m)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) / (t_m * (sin(k_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, 8.5e+27], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.5e27

   1. Initial program 31.9%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/r* 65.1%

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

   7. Simplified 65.1%

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

      1. add-sqr-sqrt 46.4%

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

      2. pow2 46.4%

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

      3. *-commutative 46.4%

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

      4. sqrt-prod 42.4%

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

      5. sqrt-prod 25.4%

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

      6. add-sqr-sqrt 46.8%

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

      7. sqrt-div 39.8%

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

      8. sqrt-pow1 41.2%

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

      9. metadata-eval 41.2%

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

   9. Applied egg-rr 41.2%

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

   if 8.5e27 < k

   1. Initial program 31.7%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in t around 0 74.7%

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

      1. associate-*r/ 74.7%

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

      2. associate-*r* 74.6%

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

      3. times-frac 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in k around 0 52.3%

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

   9. Taylor expanded in k around inf 52.3%

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

       1. associate-/r* 52.4%

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

       2. unpow2 52.4%

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

       3. unpow2 52.4%

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

       4. times-frac 55.8%

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

       5. unpow2 55.8%

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

   11. Simplified 55.8%

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

Alternative 12: 76.0% accurate, 1.3× speedup

Math

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

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 (* (pow (* (/ (sqrt 2.0) k_m) (/ l (sin k_m))) 2.0) (/ 1.0 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(((sqrt(2.0) / k_m) * (l / sin(k_m))), 2.0) * (1.0 / 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 * ((((sqrt(2.0d0) / k_m) * (l / sin(k_m))) ** 2.0d0) * (1.0d0 / 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(((Math.sqrt(2.0) / k_m) * (l / Math.sin(k_m))), 2.0) * (1.0 / 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(((math.sqrt(2.0) / k_m) * (l / math.sin(k_m))), 2.0) * (1.0 / 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(Float64(sqrt(2.0) / k_m) * Float64(l / sin(k_m))) ^ 2.0) * Float64(1.0 / 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 * ((((sqrt(2.0) / k_m) * (l / sin(k_m))) ^ 2.0) * (1.0 / 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[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

4. Applied egg-rr 29.5%

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

5. Taylor expanded in k around inf 49.5%

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

   1. times-frac 51.7%

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

7. Simplified 51.7%

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

   1. *-commutative 51.7%

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

   2. unpow-prod-down 46.6%

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

   3. pow2 46.6%

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

   4. add-sqr-sqrt 90.4%

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

   5. frac-times 89.2%

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

9. Applied egg-rr 89.2%

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

    1. *-commutative 89.2%

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

    2. times-frac 90.4%

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

11. Simplified 90.4%

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

12. Taylor expanded in k around 0 72.8%

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

13. Final simplification 72.8%

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

Alternative 13: 74.3% accurate, 1.4× speedup

Math

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

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 (pow (* l (/ (sqrt (/ 2.0 t_m)) (pow k_m 2.0))) 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 * pow((l * (sqrt((2.0 / t_m)) / pow(k_m, 2.0))), 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 * (sqrt((2.0d0 / t_m)) / (k_m ** 2.0d0))) ** 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 * Math.pow((l * (Math.sqrt((2.0 / t_m)) / Math.pow(k_m, 2.0))), 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 * math.pow((l * (math.sqrt((2.0 / t_m)) / math.pow(k_m, 2.0))), 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(l * Float64(sqrt(Float64(2.0 / t_m)) / (k_m ^ 2.0))) ^ 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 * (sqrt((2.0 / t_m)) / (k_m ^ 2.0))) ^ 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[Power[N[(l * N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 62.1%

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

7. Simplified 62.1%

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

   1. add-sqr-sqrt 47.0%

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

   2. pow2 47.0%

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

   3. *-commutative 47.0%

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

   4. sqrt-prod 43.1%

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

   5. sqrt-prod 26.0%

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

   6. add-sqr-sqrt 47.0%

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

   7. sqrt-div 36.6%

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

   8. sqrt-pow1 37.8%

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

   9. metadata-eval 37.8%

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

9. Applied egg-rr 37.8%

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

Alternative 14: 65.4% accurate, 2.0× speedup

Math

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

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 (* (* k_m k_m) (/ (* t_m (pow k_m 2.0)) (pow l 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 / ((k_m * k_m) * ((t_m * pow(k_m, 2.0)) / pow(l, 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 / ((k_m * k_m) * ((t_m * (k_m ** 2.0d0)) / (l ** 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 / ((k_m * k_m) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 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 / ((k_m * k_m) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 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(k_m * k_m) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 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 / ((k_m * k_m) * ((t_m * (k_m ^ 2.0)) / (l ^ 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[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

3. Taylor expanded in t around 0 74.4%

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

   1. associate-/l* 74.6%

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

5. Simplified 74.6%

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

   1. unpow2 74.6%

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

7. Applied egg-rr 74.6%

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

8. Taylor expanded in k around 0 65.0%

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

9. Final simplification 65.0%

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

Alternative 15: 64.1% accurate, 2.0× speedup

Math

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

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 (* (* l l) (/ 2.0 (* (pow k_m 2.0) (* 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) * (2.0 / (pow(k_m, 2.0) * (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) * (2.0d0 / ((k_m ** 2.0d0) * (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) * (2.0 / (Math.pow(k_m, 2.0) * (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) * (2.0 / (math.pow(k_m, 2.0) * (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(2.0 / Float64((k_m ^ 2.0) * 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) * (2.0 / ((k_m ^ 2.0) * (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[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

3. Simplified 38.5%

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

5. Taylor expanded in t around 0 74.5%

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

   1. associate-/l* 74.5%

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

7. Simplified 74.5%

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

8. Taylor expanded in k around 0 63.9%

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

9. Final simplification 63.9%

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

Alternative 16: 62.4% accurate, 3.8× speedup

Math

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

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) (/ (* 2.0 (pow k_m -4.0)) 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) * ((2.0 * pow(k_m, -4.0)) / 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) * ((2.0d0 * (k_m ** (-4.0d0))) / 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) * ((2.0 * Math.pow(k_m, -4.0)) / 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) * ((2.0 * math.pow(k_m, -4.0)) / 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(Float64(2.0 * (k_m ^ -4.0)) / 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) * ((2.0 * (k_m ^ -4.0)) / 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[(N[(2.0 * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 62.1%

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

7. Simplified 62.1%

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

   1. div-inv 62.1%

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

   2. pow-flip 62.1%

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

   3. metadata-eval 62.1%

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

9. Applied egg-rr 62.1%

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

    1. associate-*l/ 62.1%

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

11. Applied egg-rr 62.1%

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

12. Final simplification 62.1%

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

Alternative 17: 62.4% accurate, 3.8× speedup

Math

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

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 (* (* l l) (* (/ 2.0 t_m) (pow k_m -4.0)))))

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) * ((2.0 / t_m) * pow(k_m, -4.0)));
}

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) * ((2.0d0 / t_m) * (k_m ** (-4.0d0))))
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) * ((2.0 / t_m) * Math.pow(k_m, -4.0)));
}

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) * ((2.0 / t_m) * math.pow(k_m, -4.0)))

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(Float64(2.0 / t_m) * (k_m ^ -4.0))))
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) * ((2.0 / t_m) * (k_m ^ -4.0)));
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[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

3. Simplified 38.5%

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

5. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 62.1%

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

7. Simplified 62.1%

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

   1. div-inv 62.1%

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

   2. pow-flip 62.1%

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

   3. metadata-eval 62.1%

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

9. Applied egg-rr 62.1%

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

10. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2024180 
(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))))