Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 95.7%

Time: 15.6s

Alternatives: 13

Speedup: 3.7×

• 39.1% 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: 35.9% 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: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\sqrt{t\_2} \cdot \frac{\frac{\sqrt{2} \cdot \left(-\ell\right)}{k\_m}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot {\left(\frac{\ell \cdot \frac{\sqrt{2}}{k\_m}}{\sin k\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 6.8e-14)
      (pow (* (sqrt t_2) (/ (/ (* (sqrt 2.0) (- l)) k_m) (sin k_m))) 2.0)
      (* t_2 (pow (/ (* l (/ (sqrt 2.0) k_m)) (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 <= 6.8e-14) {
		tmp = pow((sqrt(t_2) * (((sqrt(2.0) * -l) / k_m) / sin(k_m))), 2.0);
	} else {
		tmp = t_2 * pow(((l * (sqrt(2.0) / k_m)) / 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 <= 6.8d-14) then
        tmp = (sqrt(t_2) * (((sqrt(2.0d0) * -l) / k_m) / sin(k_m))) ** 2.0d0
    else
        tmp = t_2 * (((l * (sqrt(2.0d0) / k_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 t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 6.8e-14) {
		tmp = Math.pow((Math.sqrt(t_2) * (((Math.sqrt(2.0) * -l) / k_m) / Math.sin(k_m))), 2.0);
	} else {
		tmp = t_2 * Math.pow(((l * (Math.sqrt(2.0) / k_m)) / 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 <= 6.8e-14:
		tmp = math.pow((math.sqrt(t_2) * (((math.sqrt(2.0) * -l) / k_m) / math.sin(k_m))), 2.0)
	else:
		tmp = t_2 * math.pow(((l * (math.sqrt(2.0) / k_m)) / 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 <= 6.8e-14)
		tmp = Float64(sqrt(t_2) * Float64(Float64(Float64(sqrt(2.0) * Float64(-l)) / k_m) / sin(k_m))) ^ 2.0;
	else
		tmp = Float64(t_2 * (Float64(Float64(l * Float64(sqrt(2.0) / k_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)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 6.8e-14)
		tmp = (sqrt(t_2) * (((sqrt(2.0) * -l) / k_m) / sin(k_m))) ^ 2.0;
	else
		tmp = t_2 * (((l * (sqrt(2.0) / k_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_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6.8e-14], N[Power[N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-l)), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(t$95$2 * N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000006e-14

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

      1. add-sqr-sqrt 28.7%

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

      2. pow2 28.7%

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

   6. Applied egg-rr 29.9%

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

      1. associate-/r* 31.1%

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

   8. Simplified 31.1%

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

   9. Taylor expanded in l around 0 47.4%

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

       1. associate-/l* 47.5%

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

       2. associate-/r* 47.5%

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

   11. Simplified 47.5%

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

   12. Taylor expanded in t around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 48.5%

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

   14. Simplified 48.5%

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

   if 6.80000000000000006e-14 < k

   1. Initial program 27.4%

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

   3. Simplified 45.6%

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

      1. add-sqr-sqrt 42.8%

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

      2. pow2 42.8%

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

   6. Applied egg-rr 18.7%

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

      1. associate-/r* 18.7%

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

   8. Simplified 18.7%

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

   9. Taylor expanded in l around 0 43.3%

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

       1. associate-/l* 43.3%

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

       2. associate-/r* 43.4%

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

   11. Simplified 43.4%

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

       1. unpow-prod-down 40.8%

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

       2. pow2 40.8%

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

       3. add-sqr-sqrt 96.3%

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

   13. Applied egg-rr 96.3%

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

       1. associate-*r/ 96.2%

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

   15. Applied egg-rr 96.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\frac{\sqrt{2} \cdot \left(-\ell\right)}{k}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell \cdot \frac{\sqrt{2}}{k}}{\sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell \cdot \frac{\sqrt{2}}{k\_m}}{\sin k\_m}\\ t_3 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\sqrt{t\_3} \cdot t\_2\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot {t\_2}^{2}\\ \end{array} \end{array} \end{array} \]

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 (/ (* l (/ (sqrt 2.0) k_m)) (sin k_m))) (t_3 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 6.8e-14) (pow (* (sqrt t_3) t_2) 2.0) (* t_3 (pow t_2 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 = (l * (sqrt(2.0) / k_m)) / sin(k_m);
	double t_3 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 6.8e-14) {
		tmp = pow((sqrt(t_3) * t_2), 2.0);
	} else {
		tmp = t_3 * pow(t_2, 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 = (l * (sqrt(2.0d0) / k_m)) / sin(k_m)
    t_3 = cos(k_m) / t_m
    if (k_m <= 6.8d-14) then
        tmp = (sqrt(t_3) * t_2) ** 2.0d0
    else
        tmp = t_3 * (t_2 ** 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 = (l * (Math.sqrt(2.0) / k_m)) / Math.sin(k_m);
	double t_3 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 6.8e-14) {
		tmp = Math.pow((Math.sqrt(t_3) * t_2), 2.0);
	} else {
		tmp = t_3 * Math.pow(t_2, 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 = (l * (math.sqrt(2.0) / k_m)) / math.sin(k_m)
	t_3 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 6.8e-14:
		tmp = math.pow((math.sqrt(t_3) * t_2), 2.0)
	else:
		tmp = t_3 * math.pow(t_2, 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(Float64(l * Float64(sqrt(2.0) / k_m)) / sin(k_m))
	t_3 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 6.8e-14)
		tmp = Float64(sqrt(t_3) * t_2) ^ 2.0;
	else
		tmp = Float64(t_3 * (t_2 ^ 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 = (l * (sqrt(2.0) / k_m)) / sin(k_m);
	t_3 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 6.8e-14)
		tmp = (sqrt(t_3) * t_2) ^ 2.0;
	else
		tmp = t_3 * (t_2 ^ 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[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $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, 6.8e-14], N[Power[N[(N[Sqrt[t$95$3], $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision], N[(t$95$3 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000006e-14

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

      1. add-sqr-sqrt 28.7%

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

      2. pow2 28.7%

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

   6. Applied egg-rr 29.9%

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

      1. associate-/r* 31.1%

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

   8. Simplified 31.1%

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

   9. Taylor expanded in l around 0 47.4%

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

       1. associate-/l* 47.5%

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

       2. associate-/r* 47.5%

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

   11. Simplified 47.5%

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

       1. associate-*r/ 95.1%

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

   13. Applied egg-rr 48.5%

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

   if 6.80000000000000006e-14 < k

   1. Initial program 27.4%

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

   3. Simplified 45.6%

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

      1. add-sqr-sqrt 42.8%

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

      2. pow2 42.8%

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

   6. Applied egg-rr 18.7%

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

      1. associate-/r* 18.7%

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

   8. Simplified 18.7%

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

   9. Taylor expanded in l around 0 43.3%

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

       1. associate-/l* 43.3%

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

       2. associate-/r* 43.4%

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

   11. Simplified 43.4%

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

       1. unpow-prod-down 40.8%

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

       2. pow2 40.8%

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

       3. add-sqr-sqrt 96.3%

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

   13. Applied egg-rr 96.3%

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

       1. associate-*r/ 96.2%

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

   15. Applied egg-rr 96.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \frac{\sqrt{2}}{k}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell \cdot \frac{\sqrt{2}}{k}}{\sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\frac{\ell \cdot \frac{\sqrt{2}}{k\_m}}{\sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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 6.8e-14)
    (pow (/ (* (/ (* l (sqrt 2.0)) k_m) (sqrt (/ 1.0 t_m))) (sin k_m)) 2.0)
    (* (/ (cos k_m) t_m) (pow (/ (* l (/ (sqrt 2.0) k_m)) (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 <= 6.8e-14) {
		tmp = pow(((((l * sqrt(2.0)) / k_m) * sqrt((1.0 / t_m))) / sin(k_m)), 2.0);
	} else {
		tmp = (cos(k_m) / t_m) * pow(((l * (sqrt(2.0) / k_m)) / 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 <= 6.8d-14) then
        tmp = ((((l * sqrt(2.0d0)) / k_m) * sqrt((1.0d0 / t_m))) / sin(k_m)) ** 2.0d0
    else
        tmp = (cos(k_m) / t_m) * (((l * (sqrt(2.0d0) / k_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 <= 6.8e-14) {
		tmp = Math.pow(((((l * Math.sqrt(2.0)) / k_m) * Math.sqrt((1.0 / t_m))) / Math.sin(k_m)), 2.0);
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow(((l * (Math.sqrt(2.0) / k_m)) / 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 <= 6.8e-14:
		tmp = math.pow(((((l * math.sqrt(2.0)) / k_m) * math.sqrt((1.0 / t_m))) / math.sin(k_m)), 2.0)
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow(((l * (math.sqrt(2.0) / k_m)) / 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 <= 6.8e-14)
		tmp = Float64(Float64(Float64(Float64(l * sqrt(2.0)) / k_m) * sqrt(Float64(1.0 / t_m))) / sin(k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(Float64(l * Float64(sqrt(2.0) / k_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 <= 6.8e-14)
		tmp = ((((l * sqrt(2.0)) / k_m) * sqrt((1.0 / t_m))) / sin(k_m)) ^ 2.0;
	else
		tmp = (cos(k_m) / t_m) * (((l * (sqrt(2.0) / k_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, 6.8e-14], N[Power[N[(N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000006e-14

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

      1. add-sqr-sqrt 28.7%

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

      2. pow2 28.7%

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

   6. Applied egg-rr 29.9%

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

      1. associate-/r* 31.1%

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

   8. Simplified 31.1%

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

   9. Taylor expanded in l around 0 47.4%

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

       1. associate-/l* 47.5%

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

       2. associate-/r* 47.5%

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

   11. Simplified 47.5%

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

   12. Taylor expanded in l around 0 47.4%

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

       1. associate-/r* 48.5%

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

       2. associate-*r/ 48.5%

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

       3. associate-*l/ 48.5%

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

   14. Simplified 48.5%

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

   15. Taylor expanded in k around 0 45.7%

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

   if 6.80000000000000006e-14 < k

   1. Initial program 27.4%

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

   3. Simplified 45.6%

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

      1. add-sqr-sqrt 42.8%

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

      2. pow2 42.8%

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

   6. Applied egg-rr 18.7%

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

      1. associate-/r* 18.7%

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

   8. Simplified 18.7%

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

   9. Taylor expanded in l around 0 43.3%

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

       1. associate-/l* 43.3%

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

       2. associate-/r* 43.4%

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

   11. Simplified 43.4%

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

       1. unpow-prod-down 40.8%

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

       2. pow2 40.8%

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

       3. add-sqr-sqrt 96.3%

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

   13. Applied egg-rr 96.3%

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

       1. associate-*r/ 96.2%

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

   15. Applied egg-rr 96.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k} \cdot \sqrt{\frac{1}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell \cdot \frac{\sqrt{2}}{k}}{\sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (/ (* l (/ (sqrt 2.0) k_m)) (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) {
	return t_s * ((cos(k_m) / t_m) * pow(((l * (sqrt(2.0) / k_m)) / sin(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 * ((cos(k_m) / t_m) * (((l * (sqrt(2.0d0) / k_m)) / sin(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 * ((Math.cos(k_m) / t_m) * Math.pow(((l * (Math.sqrt(2.0) / k_m)) / Math.sin(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 * ((math.cos(k_m) / t_m) * math.pow(((l * (math.sqrt(2.0) / k_m)) / math.sin(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(cos(k_m) / t_m) * (Float64(Float64(l * Float64(sqrt(2.0) / k_m)) / sin(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 * ((cos(k_m) / t_m) * (((l * (sqrt(2.0) / k_m)) / sin(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[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

3. Simplified 44.7%

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

   1. add-sqr-sqrt 32.6%

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

   2. pow2 32.6%

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

6. Applied egg-rr 26.8%

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

   1. associate-/r* 27.7%

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

8. Simplified 27.7%

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

9. Taylor expanded in l around 0 46.3%

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

    1. associate-/l* 46.3%

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

    2. associate-/r* 46.3%

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

11. Simplified 46.3%

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

    1. unpow-prod-down 44.9%

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

    2. pow2 44.9%

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

    3. add-sqr-sqrt 94.7%

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

13. Applied egg-rr 94.7%

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

    1. associate-*r/ 95.4%

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

15. Applied egg-rr 95.4%

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

16. Final simplification 95.4%

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

Alternative 5: 93.1% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 35.2%

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

3. Simplified 44.7%

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

   1. add-sqr-sqrt 32.6%

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

   2. pow2 32.6%

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

6. Applied egg-rr 26.8%

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

   1. associate-/r* 27.7%

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

8. Simplified 27.7%

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

9. Taylor expanded in l around 0 46.3%

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

    1. associate-/l* 46.3%

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

    2. associate-/r* 46.3%

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

11. Simplified 46.3%

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

    1. unpow-prod-down 44.9%

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

    2. pow2 44.9%

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

    3. add-sqr-sqrt 94.7%

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

13. Applied egg-rr 94.7%

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

    1. associate-*r/ 94.6%

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

    2. associate-/l/ 94.6%

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

15. Applied egg-rr 94.6%

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

    1. *-commutative 94.6%

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

    2. associate-/l* 94.7%

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

    3. associate-*r/ 94.6%

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

    4. associate-/l/ 95.4%

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

    5. *-commutative 95.4%

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

    6. associate-/l* 95.4%

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

17. Simplified 95.4%

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

Alternative 6: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{2}}{k\_m}}{k\_m \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(t\_m \cdot \frac{0.5 - 0.5 \cdot \cos \left(k\_m \cdot 2\right)}{\cos k\_m \cdot {\ell}^{2}}\right)}\\ \end{array} \end{array} \]

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 7e-8)
    (pow (* l (/ (/ (sqrt 2.0) k_m) (* k_m (sqrt t_m)))) 2.0)
    (/
     2.0
     (*
      (* k_m k_m)
      (*
       t_m
       (/ (- 0.5 (* 0.5 (cos (* 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 (k_m <= 7e-8) {
		tmp = pow((l * ((sqrt(2.0) / k_m) / (k_m * sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 / ((k_m * k_m) * (t_m * ((0.5 - (0.5 * cos((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 (k_m <= 7d-8) then
        tmp = (l * ((sqrt(2.0d0) / k_m) / (k_m * sqrt(t_m)))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k_m * k_m) * (t_m * ((0.5d0 - (0.5d0 * cos((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 (k_m <= 7e-8) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / k_m) / (k_m * Math.sqrt(t_m)))), 2.0);
	} else {
		tmp = 2.0 / ((k_m * k_m) * (t_m * ((0.5 - (0.5 * Math.cos((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 k_m <= 7e-8:
		tmp = math.pow((l * ((math.sqrt(2.0) / k_m) / (k_m * math.sqrt(t_m)))), 2.0)
	else:
		tmp = 2.0 / ((k_m * k_m) * (t_m * ((0.5 - (0.5 * math.cos((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 (k_m <= 7e-8)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / k_m) / Float64(k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(t_m * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(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 (k_m <= 7e-8)
		tmp = (l * ((sqrt(2.0) / k_m) / (k_m * sqrt(t_m)))) ^ 2.0;
	else
		tmp = 2.0 / ((k_m * k_m) * (t_m * ((0.5 - (0.5 * cos((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[k$95$m, 7e-8], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.00000000000000048e-8

   1. Initial program 38.5%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in k around 0 58.3%

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

   6. Taylor expanded in k around inf 69.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 69.4%

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

      2. metadata-eval 69.4%

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

      3. *-commutative 69.4%

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

      4. associate-*r/ 69.4%

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

      5. metadata-eval 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in k around 0 70.2%

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

       1. associate-/r* 70.2%

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

   11. Simplified 70.2%

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

       1. pow1 70.2%

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

   13. Applied egg-rr 44.1%

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

       1. unpow1 44.1%

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

   15. Simplified 44.1%

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

   if 7.00000000000000048e-8 < k

   1. Initial program 26.3%

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

   3. Taylor expanded in t around 0 70.6%

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

      1. associate-/l* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 71.7%

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

   9. Applied egg-rr 71.7%

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

       1. div-sub 71.7%

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

       2. +-inverses 71.7%

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

       3. cos-0 71.7%

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

       4. metadata-eval 71.7%

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

       5. count-2 71.7%

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

       6. *-commutative 71.7%

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

   11. Simplified 71.7%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
   12. Step-by-step derivation

       1. associate-/l* 71.6%

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

       2. div-inv 71.6%

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

       3. *-commutative 71.6%

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

       4. metadata-eval 71.6%

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

       5. pow2 71.6%

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

       6. *-commutative 71.6%

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

       7. pow2 71.6%

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

   13. Applied egg-rr 71.6%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-8}:\\ \;\;\;\;{\left(\ell \cdot \frac{\frac{\sqrt{2}}{k}}{k \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{0.5 - 0.5 \cdot \cos \left(k \cdot 2\right)}{\cos k \cdot {\ell}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 1.4× speedup

Math

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

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) k_m) (* k_m (sqrt t_m)))) 2.0)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * pow((l * ((sqrt(2.0) / k_m) / (k_m * sqrt(t_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 * ((sqrt(2.0d0) / k_m) / (k_m * sqrt(t_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 * Math.pow((l * ((Math.sqrt(2.0) / k_m) / (k_m * Math.sqrt(t_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 * math.pow((l * ((math.sqrt(2.0) / k_m) / (k_m * math.sqrt(t_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(l * Float64(Float64(sqrt(2.0) / k_m) / Float64(k_m * sqrt(t_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 * ((sqrt(2.0) / k_m) / (k_m * sqrt(t_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[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

3. Simplified 44.7%

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

5. Taylor expanded in k around 0 48.6%

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

6. Taylor expanded in k around inf 66.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
7. Step-by-step derivation

   1. associate-*r/ 66.3%

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

   2. metadata-eval 66.3%

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

   3. *-commutative 66.3%

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

   4. associate-*r/ 66.3%

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

   5. metadata-eval 66.3%

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

8. Simplified 66.3%

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

9. Taylor expanded in k around 0 65.3%

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

    1. associate-/r* 65.3%

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

11. Simplified 65.3%

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

    1. pow1 65.3%

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

13. Applied egg-rr 39.2%

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

    1. unpow1 39.2%

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

15. Simplified 39.2%

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

Alternative 8: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{{k\_m}^{2}} + -0.3333333333333333\right) \cdot {k\_m}^{-2}}{t\_m} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

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 1.95e+21)
    (/ 2.0 (* (* k_m k_m) (/ (* t_m (pow k_m 2.0)) (pow l 2.0))))
    (*
     (/ (* (+ (/ 2.0 (pow k_m 2.0)) -0.3333333333333333) (pow k_m -2.0)) t_m)
     (* 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 <= 1.95e+21) {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * pow(k_m, 2.0)) / pow(l, 2.0)));
	} else {
		tmp = ((((2.0 / pow(k_m, 2.0)) + -0.3333333333333333) * pow(k_m, -2.0)) / t_m) * (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 <= 1.95d+21) then
        tmp = 2.0d0 / ((k_m * k_m) * ((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)))
    else
        tmp = ((((2.0d0 / (k_m ** 2.0d0)) + (-0.3333333333333333d0)) * (k_m ** (-2.0d0))) / t_m) * (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 <= 1.95e+21) {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)));
	} else {
		tmp = ((((2.0 / Math.pow(k_m, 2.0)) + -0.3333333333333333) * Math.pow(k_m, -2.0)) / t_m) * (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 <= 1.95e+21:
		tmp = 2.0 / ((k_m * k_m) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)))
	else:
		tmp = ((((2.0 / math.pow(k_m, 2.0)) + -0.3333333333333333) * math.pow(k_m, -2.0)) / t_m) * (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 <= 1.95e+21)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 / (k_m ^ 2.0)) + -0.3333333333333333) * (k_m ^ -2.0)) / t_m) * 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 <= 1.95e+21)
		tmp = 2.0 / ((k_m * k_m) * ((t_m * (k_m ^ 2.0)) / (l ^ 2.0)));
	else
		tmp = ((((2.0 / (k_m ^ 2.0)) + -0.3333333333333333) * (k_m ^ -2.0)) / t_m) * (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, 1.95e+21], 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], N[(N[(N[(N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.95e21

   1. Initial program 38.9%

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

   3. Taylor expanded in t around 0 77.7%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

      1. unpow2 78.6%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in k around 0 71.0%

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

   if 1.95e21 < k

   1. Initial program 24.1%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 18.5%

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

   6. Taylor expanded in k around inf 57.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 57.2%

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

      2. metadata-eval 57.2%

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

      3. *-commutative 57.2%

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

      4. associate-*r/ 57.2%

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

      5. metadata-eval 57.2%

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

   8. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot {k}^{2}} - \frac{0.3333333333333333}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. div-inv 57.2%

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

      2. *-commutative 57.2%

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

      3. pow-flip 58.7%

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

      4. metadata-eval 58.7%

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

   10. Applied egg-rr 58.7%

       \[\leadsto \color{blue}{\left(\left(\frac{2}{{k}^{2} \cdot t} - \frac{0.3333333333333333}{t}\right) \cdot {k}^{-2}\right)} \cdot \left(\ell \cdot \ell\right) \]
   11. Step-by-step derivation

       1. associate-/r* 58.7%

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

       2. metadata-eval 58.7%

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

       3. associate-*r/ 58.7%

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

       4. div-sub 58.7%

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

       5. associate-*l/ 58.7%

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

       6. sub-neg 58.7%

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

       7. associate-*r/ 58.7%

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

       8. metadata-eval 58.7%

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

       9. metadata-eval 58.7%

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

   12. Simplified 58.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{2}{{k}^{2}} + -0.3333333333333333\right) \cdot {k}^{-2}}{t}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{{k}^{2}} + -0.3333333333333333\right) \cdot {k}^{-2}}{t} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{{\left(k\_m \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.95e+21)
    (/ 2.0 (* (* k_m k_m) (/ (* t_m (pow k_m 2.0)) (pow l 2.0))))
    (* (* l l) (/ -0.3333333333333333 (pow (* k_m (sqrt t_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e+21) {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * pow(k_m, 2.0)) / pow(l, 2.0)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / pow((k_m * sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.95d+21) then
        tmp = 2.0d0 / ((k_m * k_m) * ((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / ((k_m * sqrt(t_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e+21) {
		tmp = 2.0 / ((k_m * k_m) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / Math.pow((k_m * Math.sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.95e+21:
		tmp = 2.0 / ((k_m * k_m) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)))
	else:
		tmp = (l * l) * (-0.3333333333333333 / math.pow((k_m * math.sqrt(t_m)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e+21)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / (Float64(k_m * sqrt(t_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.95e+21)
		tmp = 2.0 / ((k_m * k_m) * ((t_m * (k_m ^ 2.0)) / (l ^ 2.0)));
	else
		tmp = (l * l) * (-0.3333333333333333 / ((k_m * sqrt(t_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.95e+21], 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], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[Power[N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.95e21

   1. Initial program 38.9%

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

   3. Taylor expanded in t around 0 77.7%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

      1. unpow2 78.6%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in k around 0 71.0%

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

   if 1.95e21 < k

   1. Initial program 24.1%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 18.5%

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

   6. Taylor expanded in k around inf 57.2%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. unpow2 71.2%

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

   8. Applied egg-rr 57.2%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 29.3%

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

      2. pow2 29.3%

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

      3. sqrt-prod 29.3%

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

      4. sqrt-prod 29.3%

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

      5. add-sqr-sqrt 29.3%

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

   10. Applied egg-rr 29.3%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{{\left(k \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(t\_m \cdot \frac{{k\_m}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{{\left(k\_m \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.95e+21)
    (/ 2.0 (* (* k_m k_m) (* t_m (/ (pow k_m 2.0) (pow l 2.0)))))
    (* (* l l) (/ -0.3333333333333333 (pow (* k_m (sqrt t_m)) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e+21) {
		tmp = 2.0 / ((k_m * k_m) * (t_m * (pow(k_m, 2.0) / pow(l, 2.0))));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / pow((k_m * sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.95d+21) then
        tmp = 2.0d0 / ((k_m * k_m) * (t_m * ((k_m ** 2.0d0) / (l ** 2.0d0))))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / ((k_m * sqrt(t_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e+21) {
		tmp = 2.0 / ((k_m * k_m) * (t_m * (Math.pow(k_m, 2.0) / Math.pow(l, 2.0))));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / Math.pow((k_m * Math.sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.95e+21:
		tmp = 2.0 / ((k_m * k_m) * (t_m * (math.pow(k_m, 2.0) / math.pow(l, 2.0))))
	else:
		tmp = (l * l) * (-0.3333333333333333 / math.pow((k_m * math.sqrt(t_m)), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e+21)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(t_m * Float64((k_m ^ 2.0) / (l ^ 2.0)))));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / (Float64(k_m * sqrt(t_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.95e+21)
		tmp = 2.0 / ((k_m * k_m) * (t_m * ((k_m ^ 2.0) / (l ^ 2.0))));
	else
		tmp = (l * l) * (-0.3333333333333333 / ((k_m * sqrt(t_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.95e+21], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$m * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[Power[N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.95e21

   1. Initial program 38.9%

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

   3. Taylor expanded in t around 0 77.7%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

      1. unpow2 78.6%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in k around 0 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-/l* 70.9%

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

   10. Simplified 70.9%

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

   if 1.95e21 < k

   1. Initial program 24.1%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 18.5%

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

   6. Taylor expanded in k around inf 57.2%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. unpow2 71.2%

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

   8. Applied egg-rr 57.2%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\ell \cdot \ell\right) \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 29.3%

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

      2. pow2 29.3%

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

      3. sqrt-prod 29.3%

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

      4. sqrt-prod 29.3%

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

      5. add-sqr-sqrt 29.3%

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

   10. Applied egg-rr 29.3%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{{\left(k \cdot \sqrt{t}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{{\left(k \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Simplified 44.7%

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

5. Taylor expanded in k around 0 48.6%

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

6. Taylor expanded in k around inf 66.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{1}{{k}^{2} \cdot t} - 0.3333333333333333 \cdot \frac{1}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
7. Step-by-step derivation

   1. associate-*r/ 66.3%

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

   2. metadata-eval 66.3%

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

   3. *-commutative 66.3%

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

   4. associate-*r/ 66.3%

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

   5. metadata-eval 66.3%

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

8. Simplified 66.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot {k}^{2}} - \frac{0.3333333333333333}{t}}{{k}^{2}}} \cdot \left(\ell \cdot \ell\right) \]
9. Step-by-step derivation

   1. unpow2 76.7%

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

10. Applied egg-rr 66.3%

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

11. Final simplification 66.3%

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

Alternative 12: 64.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t\_m \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

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 1.95e+21)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (* (* l l) (/ -0.3333333333333333 (* t_m (* k_m 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 <= 1.95e+21) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (t_m * (k_m * 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 <= 1.95d+21) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / (t_m * (k_m * 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 <= 1.95e+21) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (t_m * (k_m * 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 <= 1.95e+21:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = (l * l) * (-0.3333333333333333 / (t_m * (k_m * 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 <= 1.95e+21)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(t_m * Float64(k_m * 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 <= 1.95e+21)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = (l * l) * (-0.3333333333333333 / (t_m * (k_m * 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, 1.95e+21], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.95e21

   1. Initial program 38.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in k around 0 68.0%

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

   if 1.95e21 < k

   1. Initial program 24.1%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 18.5%

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

   6. Taylor expanded in k around inf 57.2%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. unpow2 71.2%

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

   8. Applied egg-rr 57.2%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+21}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.1% accurate, 38.3× speedup

Math

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

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) (/ -0.3333333333333333 (* t_m (* k_m k_m))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

3. Simplified 44.7%

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

5. Taylor expanded in k around 0 48.6%

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

6. Taylor expanded in k around inf 27.9%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
7. Step-by-step derivation

   1. unpow2 76.7%

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

8. Applied egg-rr 27.9%

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

9. Final simplification 27.9%

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

Reproduce

herbie shell --seed 2024167 
(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))))