Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 82.9%

Time: 25.3s

Alternatives: 8

Speedup: 3.8×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k_m}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 150000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k_m} \cdot \frac{\sqrt{2}}{\sin k_m}\right) \cdot \sqrt{t_2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2 \cdot {\ell}^{2}}{{\left(k_m \cdot \sin k_m\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 150000000000.0)
      (pow (* (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt t_2)) 2.0)
      (* 2.0 (/ (* t_2 (pow l 2.0)) (pow (* 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 <= 150000000000.0) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt(t_2)), 2.0);
	} else {
		tmp = 2.0 * ((t_2 * pow(l, 2.0)) / pow((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 <= 150000000000.0d0) then
        tmp = (((l / k_m) * (sqrt(2.0d0) / sin(k_m))) * sqrt(t_2)) ** 2.0d0
    else
        tmp = 2.0d0 * ((t_2 * (l ** 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 <= 150000000000.0) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt(t_2)), 2.0);
	} else {
		tmp = 2.0 * ((t_2 * Math.pow(l, 2.0)) / Math.pow((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 <= 150000000000.0:
		tmp = math.pow((((l / k_m) * (math.sqrt(2.0) / math.sin(k_m))) * math.sqrt(t_2)), 2.0)
	else:
		tmp = 2.0 * ((t_2 * math.pow(l, 2.0)) / math.pow((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 <= 150000000000.0)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(t_2)) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(t_2 * (l ^ 2.0)) / (Float64(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 <= 150000000000.0)
		tmp = (((l / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt(t_2)) ^ 2.0;
	else
		tmp = 2.0 * ((t_2 * (l ^ 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, 150000000000.0], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(t$95$2 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e11

   1. Initial program 38.2%

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

      1. associate-*l* 38.2%

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

      2. associate-/r* 37.7%

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

      3. sub-neg 37.7%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 22.0%

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

      7. sqr-neg 22.0%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.7%

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

      11. +-commutative 37.7%

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

      12. associate-+l+ 42.2%

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

   3. Simplified 42.2%

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

   6. Applied egg-rr 19.5%

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

      1. unpow2 19.5%

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

      2. associate-/r/ 20.6%

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

      3. associate-*r/ 20.6%

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

   8. Simplified 20.6%

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

   9. Taylor expanded in l around 0 37.3%

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

       1. times-frac 37.8%

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

   11. Simplified 37.8%

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

   if 1.5e11 < k

   1. Initial program 37.8%

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

      1. associate-/r* 37.9%

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

      2. associate-*l* 37.9%

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

      3. associate-*l/ 37.9%

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

      4. associate-/l* 37.9%

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

      5. +-commutative 37.9%

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

      6. unpow2 37.9%

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

      7. sqr-neg 37.9%

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

      8. distribute-frac-neg 37.9%

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

      9. distribute-frac-neg 37.9%

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

      10. unpow2 37.9%

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

      11. associate--l+ 50.3%

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

      12. metadata-eval 50.3%

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

      13. +-rgt-identity 50.3%

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

      14. unpow2 50.3%

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

      15. distribute-frac-neg 50.3%

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

      16. distribute-frac-neg 50.3%

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

      17. sqr-neg 50.3%

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

      18. unpow2 50.3%

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

   3. Simplified 50.3%

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

      1. cube-mult 50.3%

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

      2. times-frac 50.4%

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

      3. times-frac 56.8%

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

      4. pow2 56.8%

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

   6. Applied egg-rr 56.8%

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

      1. associate-/r/ 56.8%

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

      2. associate-/r/ 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in t around 0 77.1%

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

       1. times-frac 77.1%

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

       2. associate-/r* 77.1%

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

   11. Simplified 77.1%

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

       1. pow2 77.1%

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

       2. frac-times 81.6%

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

       3. pow2 81.6%

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

       4. pow-prod-down 81.7%

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

   13. Applied egg-rr 81.7%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 150000000000:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot {\ell}^{2}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k_m}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k_m}{t_m} \cdot {\ell}^{2}}{{\left(k_m \cdot \sin k_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.65e-30)
    (pow (* (/ l (/ (pow k_m 2.0) (sqrt 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (/ (* (/ (cos k_m) t_m) (pow l 2.0)) (pow (* 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 <= 2.65e-30) {
		tmp = pow(((l / (pow(k_m, 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (((cos(k_m) / t_m) * pow(l, 2.0)) / pow((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 <= 2.65d-30) then
        tmp = ((l / ((k_m ** 2.0d0) / sqrt(2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((cos(k_m) / t_m) * (l ** 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 <= 2.65e-30) {
		tmp = Math.pow(((l / (Math.pow(k_m, 2.0) / Math.sqrt(2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (((Math.cos(k_m) / t_m) * Math.pow(l, 2.0)) / Math.pow((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 <= 2.65e-30:
		tmp = math.pow(((l / (math.pow(k_m, 2.0) / math.sqrt(2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * (((math.cos(k_m) / t_m) * math.pow(l, 2.0)) / math.pow((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 <= 2.65e-30)
		tmp = Float64(Float64(l / Float64((k_m ^ 2.0) / sqrt(2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k_m) / t_m) * (l ^ 2.0)) / (Float64(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 <= 2.65e-30)
		tmp = ((l / ((k_m ^ 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((cos(k_m) / t_m) * (l ^ 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, 2.65e-30], N[Power[N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.64999999999999987e-30

   1. Initial program 39.0%

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

      1. associate-*l* 39.0%

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

      2. associate-/r* 38.5%

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

      3. sub-neg 38.5%

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

      4. distribute-rgt-in 33.0%

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

      5. unpow2 33.0%

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

      6. times-frac 22.2%

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

      7. sqr-neg 22.2%

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

      8. times-frac 33.0%

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

      9. unpow2 33.0%

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

      10. distribute-rgt-in 38.5%

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

      11. +-commutative 38.5%

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

      12. associate-+l+ 42.7%

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

   3. Simplified 42.7%

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

   6. Applied egg-rr 18.6%

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

      1. unpow2 18.6%

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

      2. associate-/r/ 19.8%

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

      3. associate-*r/ 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in k around 0 26.6%

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

       1. associate-/l* 26.6%

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

   11. Simplified 26.6%

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

   if 2.64999999999999987e-30 < k

   1. Initial program 35.7%

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

      1. associate-/r* 35.8%

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

      2. associate-*l* 35.8%

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

      3. associate-*l/ 35.8%

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

      4. associate-/l* 35.7%

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

      5. +-commutative 35.7%

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

      6. unpow2 35.7%

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

      7. sqr-neg 35.7%

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

      8. distribute-frac-neg 35.7%

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

      9. distribute-frac-neg 35.7%

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

      10. unpow2 35.7%

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

      11. associate--l+ 48.3%

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

      12. metadata-eval 48.3%

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

      13. +-rgt-identity 48.3%

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

      14. unpow2 48.3%

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

      15. distribute-frac-neg 48.3%

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

      16. distribute-frac-neg 48.3%

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

      17. sqr-neg 48.3%

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

      18. unpow2 48.3%

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

   3. Simplified 48.3%

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

      1. cube-mult 48.3%

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

      2. times-frac 48.4%

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

      3. times-frac 56.8%

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

      4. pow2 56.8%

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

   6. Applied egg-rr 56.8%

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

      1. associate-/r/ 56.8%

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

      2. associate-/r/ 56.7%

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

   8. Simplified 56.7%

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

   9. Taylor expanded in t around 0 78.0%

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

       1. times-frac 78.0%

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

       2. associate-/r* 78.0%

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

   11. Simplified 78.0%

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

       1. pow2 78.0%

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

       2. frac-times 82.1%

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

       3. pow2 82.1%

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

       4. pow-prod-down 82.2%

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

   13. Applied egg-rr 82.2%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot {\ell}^{2}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 0.0026:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k_m}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{\cos k_m \cdot \left(\ell \cdot \ell\right)}{t_m \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.0026)
    (pow (* (/ l (/ (pow k_m 2.0) (sqrt 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (/
     2.0
     (/
      (pow k_m 2.0)
      (/ (* (cos k_m) (* l l)) (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0026) {
		tmp = pow(((l / (pow(k_m, 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) / ((cos(k_m) * (l * l)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 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 <= 0.0026d0) then
        tmp = ((l / ((k_m ** 2.0d0) / sqrt(2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) / ((cos(k_m) * (l * l)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 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 <= 0.0026) {
		tmp = Math.pow(((l / (Math.pow(k_m, 2.0) / Math.sqrt(2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.cos(k_m) * (l * l)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 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 <= 0.0026:
		tmp = math.pow(((l / (math.pow(k_m, 2.0) / math.sqrt(2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) / ((math.cos(k_m) * (l * l)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 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 <= 0.0026)
		tmp = Float64(Float64(l / Float64((k_m ^ 2.0) / sqrt(2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64(cos(k_m) * Float64(l * l)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 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 <= 0.0026)
		tmp = ((l / ((k_m ^ 2.0) / sqrt(2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((k_m ^ 2.0) / ((cos(k_m) * (l * l)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 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, 0.0026], N[Power[N[(N[(l / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0025999999999999999

   1. Initial program 38.2%

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

      1. associate-*l* 38.2%

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

      2. associate-/r* 37.7%

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

      3. sub-neg 37.7%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 21.8%

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

      7. sqr-neg 21.8%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.7%

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

      11. +-commutative 37.7%

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

      12. associate-+l+ 42.3%

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

   3. Simplified 42.3%

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

   6. Applied egg-rr 19.3%

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

      1. unpow2 19.3%

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

      2. associate-/r/ 20.4%

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

      3. associate-*r/ 20.4%

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

   8. Simplified 20.4%

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

   9. Taylor expanded in k around 0 26.6%

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

       1. associate-/l* 26.6%

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

   11. Simplified 26.6%

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

   if 0.0025999999999999999 < k

   1. Initial program 37.7%

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

   3. Taylor expanded in t around 0 76.6%

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

      1. associate-/l* 80.8%

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

   5. Simplified 80.8%

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

      1. unpow2 56.5%

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

   7. Applied egg-rr 80.8%

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

      1. unpow2 80.8%

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

      2. sin-mult 80.7%

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

   9. Applied egg-rr 80.7%

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

       1. div-sub 80.7%

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

       2. +-inverses 80.7%

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

       3. cos-0 80.7%

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

       4. metadata-eval 80.7%

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

       5. count-2 80.7%

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

   11. Simplified 80.7%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0026:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.9 \cdot 10^{-120}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k_m}^{-4}\right)}{t_m}\\ \mathbf{elif}\;k_m \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{{\ell}^{2}}{t_m \cdot {k_m}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{\cos k_m \cdot \left(\ell \cdot \ell\right)}{t_m \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.9e-120)
    (* 2.0 (/ (* l (* l (pow k_m -4.0))) t_m))
    (if (<= k_m 8.5e-5)
      (/ 2.0 (/ (pow k_m 2.0) (/ (pow l 2.0) (* t_m (pow k_m 2.0)))))
      (/
       2.0
       (/
        (pow k_m 2.0)
        (/
         (* (cos k_m) (* l l))
         (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.9e-120) {
		tmp = 2.0 * ((l * (l * pow(k_m, -4.0))) / t_m);
	} else if (k_m <= 8.5e-5) {
		tmp = 2.0 / (pow(k_m, 2.0) / (pow(l, 2.0) / (t_m * pow(k_m, 2.0))));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) / ((cos(k_m) * (l * l)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 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 <= 2.9d-120) then
        tmp = 2.0d0 * ((l * (l * (k_m ** (-4.0d0)))) / t_m)
    else if (k_m <= 8.5d-5) then
        tmp = 2.0d0 / ((k_m ** 2.0d0) / ((l ** 2.0d0) / (t_m * (k_m ** 2.0d0))))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) / ((cos(k_m) * (l * l)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 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 <= 2.9e-120) {
		tmp = 2.0 * ((l * (l * Math.pow(k_m, -4.0))) / t_m);
	} else if (k_m <= 8.5e-5) {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 2.0))));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.cos(k_m) * (l * l)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 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 <= 2.9e-120:
		tmp = 2.0 * ((l * (l * math.pow(k_m, -4.0))) / t_m)
	elif k_m <= 8.5e-5:
		tmp = 2.0 / (math.pow(k_m, 2.0) / (math.pow(l, 2.0) / (t_m * math.pow(k_m, 2.0))))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) / ((math.cos(k_m) * (l * l)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 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 <= 2.9e-120)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l * (k_m ^ -4.0))) / t_m));
	elseif (k_m <= 8.5e-5)
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64(cos(k_m) * Float64(l * l)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 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 <= 2.9e-120)
		tmp = 2.0 * ((l * (l * (k_m ^ -4.0))) / t_m);
	elseif (k_m <= 8.5e-5)
		tmp = 2.0 / ((k_m ^ 2.0) / ((l ^ 2.0) / (t_m * (k_m ^ 2.0))));
	else
		tmp = 2.0 / ((k_m ^ 2.0) / ((cos(k_m) * (l * l)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 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, 2.9e-120], N[(2.0 * N[(N[(l * N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.5e-5], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.9e-120

   1. Initial program 40.3%

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

      1. associate-*l* 40.3%

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

      2. associate-/r* 39.6%

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

      3. sub-neg 39.6%

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

      4. distribute-rgt-in 33.4%

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

      5. unpow2 33.4%

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

      6. times-frac 21.7%

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

      7. sqr-neg 21.7%

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

      8. times-frac 33.4%

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

      9. unpow2 33.4%

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

      10. distribute-rgt-in 39.6%

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

      11. +-commutative 39.6%

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

      12. associate-+l+ 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in k around 0 66.9%

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

      1. unpow2 66.9%

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

   7. Applied egg-rr 66.9%

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

      1. times-frac 73.5%

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

   9. Applied egg-rr 73.5%

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

       1. associate-*r/ 74.3%

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

       2. div-inv 74.3%

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

       3. pow-flip 74.3%

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

       4. metadata-eval 74.3%

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

   11. Applied egg-rr 74.3%

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

   if 2.9e-120 < k < 8.500000000000001e-5

   1. Initial program 26.1%

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

   3. Taylor expanded in t around 0 88.5%

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

      1. associate-/l* 89.0%

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

   5. Simplified 89.0%

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

   6. Taylor expanded in k around 0 88.5%

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

   if 8.500000000000001e-5 < k

   1. Initial program 37.7%

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

   3. Taylor expanded in t around 0 76.6%

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

      1. associate-/l* 80.8%

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

   5. Simplified 80.8%

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

      1. unpow2 56.5%

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

   7. Applied egg-rr 80.8%

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

      1. unpow2 80.8%

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

      2. sin-mult 80.7%

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

   9. Applied egg-rr 80.7%

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

       1. div-sub 80.7%

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

       2. +-inverses 80.7%

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

       3. cos-0 80.7%

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

       4. metadata-eval 80.7%

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

       5. count-2 80.7%

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

   11. Simplified 80.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-120}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k_m}^{-4}\right)}{t_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{\cos k_m \cdot \left(\ell \cdot \ell\right)}{t_m \cdot {k_m}^{2}}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (* 2.0 (/ (* l (* l (pow k_m -4.0))) t_m))
    (/
     2.0
     (/ (pow k_m 2.0) (/ (* (cos k_m) (* l l)) (* t_m (pow k_m 2.0))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * ((l * (l * pow(k_m, -4.0))) / t_m);
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) / ((cos(k_m) * (l * l)) / (t_m * pow(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 ((l * l) <= 0.0d0) then
        tmp = 2.0d0 * ((l * (l * (k_m ** (-4.0d0)))) / t_m)
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) / ((cos(k_m) * (l * l)) / (t_m * (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 ((l * l) <= 0.0) {
		tmp = 2.0 * ((l * (l * Math.pow(k_m, -4.0))) / t_m);
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.cos(k_m) * (l * l)) / (t_m * Math.pow(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 (l * l) <= 0.0:
		tmp = 2.0 * ((l * (l * math.pow(k_m, -4.0))) / t_m)
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) / ((math.cos(k_m) * (l * l)) / (t_m * math.pow(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 (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l * (k_m ^ -4.0))) / t_m));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64(cos(k_m) * Float64(l * l)) / Float64(t_m * (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 ((l * l) <= 0.0)
		tmp = 2.0 * ((l * (l * (k_m ^ -4.0))) / t_m);
	else
		tmp = 2.0 / ((k_m ^ 2.0) / ((cos(k_m) * (l * l)) / (t_m * (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[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[(N[(l * N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. associate-/r* 29.3%

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

      3. sub-neg 29.3%

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

      4. distribute-rgt-in 29.3%

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

      5. unpow2 29.3%

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

      6. times-frac 25.9%

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

      7. sqr-neg 25.9%

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

      8. times-frac 29.3%

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

      9. unpow2 29.3%

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

      10. distribute-rgt-in 29.3%

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

      11. +-commutative 29.3%

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

      12. associate-+l+ 43.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. unpow2 67.4%

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

   7. Applied egg-rr 67.4%

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

      1. times-frac 88.4%

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

   9. Applied egg-rr 88.4%

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

       1. associate-*r/ 90.3%

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

       2. div-inv 90.3%

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

       3. pow-flip 90.3%

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

       4. metadata-eval 90.3%

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

   11. Applied egg-rr 90.3%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.7%

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

   3. Taylor expanded in t around 0 81.3%

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

      1. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

      1. unpow2 64.8%

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

   7. Applied egg-rr 83.4%

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

   8. Taylor expanded in k around 0 72.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k_m}^{-4}\right)}{t_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{t_m \cdot {k_m}^{2}} \cdot \frac{2}{{k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (* 2.0 (/ (* l (* l (pow k_m -4.0))) t_m))
    (* (/ (pow l 2.0) (* t_m (pow k_m 2.0))) (/ 2.0 (pow k_m 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 * ((l * (l * pow(k_m, -4.0))) / t_m);
	} else {
		tmp = (pow(l, 2.0) / (t_m * pow(k_m, 2.0))) * (2.0 / pow(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 ((l * l) <= 0.0d0) then
        tmp = 2.0d0 * ((l * (l * (k_m ** (-4.0d0)))) / t_m)
    else
        tmp = ((l ** 2.0d0) / (t_m * (k_m ** 2.0d0))) * (2.0d0 / (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 ((l * l) <= 0.0) {
		tmp = 2.0 * ((l * (l * Math.pow(k_m, -4.0))) / t_m);
	} else {
		tmp = (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 2.0))) * (2.0 / Math.pow(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 (l * l) <= 0.0:
		tmp = 2.0 * ((l * (l * math.pow(k_m, -4.0))) / t_m)
	else:
		tmp = (math.pow(l, 2.0) / (t_m * math.pow(k_m, 2.0))) * (2.0 / math.pow(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 (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l * (k_m ^ -4.0))) / t_m));
	else
		tmp = Float64(Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 2.0))) * Float64(2.0 / (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 ((l * l) <= 0.0)
		tmp = 2.0 * ((l * (l * (k_m ^ -4.0))) / t_m);
	else
		tmp = ((l ^ 2.0) / (t_m * (k_m ^ 2.0))) * (2.0 / (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[N[(l * l), $MachinePrecision], 0.0], N[(2.0 * N[(N[(l * N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 29.3%

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

      1. associate-*l* 29.3%

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

      2. associate-/r* 29.3%

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

      3. sub-neg 29.3%

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

      4. distribute-rgt-in 29.3%

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

      5. unpow2 29.3%

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

      6. times-frac 25.9%

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

      7. sqr-neg 25.9%

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

      8. times-frac 29.3%

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

      9. unpow2 29.3%

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

      10. distribute-rgt-in 29.3%

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

      11. +-commutative 29.3%

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

      12. associate-+l+ 43.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. unpow2 67.4%

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

   7. Applied egg-rr 67.4%

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

      1. times-frac 88.4%

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

   9. Applied egg-rr 88.4%

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

       1. associate-*r/ 90.3%

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

       2. div-inv 90.3%

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

       3. pow-flip 90.3%

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

       4. metadata-eval 90.3%

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

   11. Applied egg-rr 90.3%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.7%

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

   3. Taylor expanded in t around 0 81.3%

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

      1. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

   6. Taylor expanded in k around 0 69.6%

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

      1. associate-/r* 68.3%

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

   8. Simplified 68.3%

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

      1. expm1-log1p-u 34.5%

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

      2. expm1-udef 32.1%

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

      3. associate-/r/ 32.1%

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

      4. pow2 32.1%

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

      5. associate-/l/ 32.0%

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

      6. pow2 32.0%

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

   10. Applied egg-rr 32.0%

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

       1. expm1-def 34.4%

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

       2. expm1-log1p 69.6%

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

       3. *-commutative 69.6%

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

       4. *-commutative 69.6%

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

   12. Simplified 69.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{2}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 38.1%

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

   1. associate-*l* 38.1%

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

   2. associate-/r* 37.7%

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

   3. sub-neg 37.7%

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

   4. distribute-rgt-in 33.8%

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

   5. unpow2 33.8%

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

   6. times-frac 25.2%

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

   7. sqr-neg 25.2%

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

   8. times-frac 33.8%

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

   9. unpow2 33.8%

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

   10. distribute-rgt-in 37.7%

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

   11. +-commutative 37.7%

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

   12. associate-+l+ 44.3%

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

3. Simplified 44.3%

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

5. Taylor expanded in k around 0 65.4%

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

   1. unpow2 65.4%

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

7. Applied egg-rr 65.4%

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

   1. times-frac 70.4%

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

9. Applied egg-rr 70.4%

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

10. Final simplification 70.4%

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

Alternative 8: 65.7% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (* l (pow k_m -4.0)) (/ t_m l)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. associate-*l* 38.1%

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

   2. associate-/r* 37.7%

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

   3. sub-neg 37.7%

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

   4. distribute-rgt-in 33.8%

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

   5. unpow2 33.8%

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

   6. times-frac 25.2%

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

   7. sqr-neg 25.2%

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

   8. times-frac 33.8%

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

   9. unpow2 33.8%

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

   10. distribute-rgt-in 37.7%

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

   11. +-commutative 37.7%

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

   12. associate-+l+ 44.3%

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

3. Simplified 44.3%

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

5. Taylor expanded in k around 0 65.4%

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

   1. unpow2 65.4%

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

7. Applied egg-rr 65.4%

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

   1. times-frac 70.4%

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

9. Applied egg-rr 70.4%

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

    1. associate-*r/ 69.8%

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

    2. div-inv 69.8%

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

    3. pow-flip 69.8%

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

    4. metadata-eval 69.8%

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

11. Applied egg-rr 69.8%

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

    1. associate-/l* 70.4%

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

13. Simplified 70.4%

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

14. Final simplification 70.4%

    \[\leadsto 2 \cdot \frac{\ell \cdot {k}^{-4}}{\frac{t}{\ell}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))