Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 97.9%

Time: 16.1s

Alternatives: 14

Speedup: 11.6×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% 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: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 18000:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot k\right) \cdot \left(\tan k \cdot \frac{k \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(\left(t\_2 \cdot \tan k\right) \cdot t\_m\right)\right) \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 18000.0)
      (/ 2.0 (* (* t_2 k) (* (tan k) (/ (* k t_m) l))))
      (/ 2.0 (* (* (/ k l) (* (* t_2 (tan k)) t_m)) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 18000.0) {
		tmp = 2.0 / ((t_2 * k) * (tan(k) * ((k * t_m) / l)));
	} else {
		tmp = 2.0 / (((k / l) * ((t_2 * tan(k)) * t_m)) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) / l
    if (t_m <= 18000.0d0) then
        tmp = 2.0d0 / ((t_2 * k) * (tan(k) * ((k * t_m) / l)))
    else
        tmp = 2.0d0 / (((k / l) * ((t_2 * tan(k)) * t_m)) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) / l;
	double tmp;
	if (t_m <= 18000.0) {
		tmp = 2.0 / ((t_2 * k) * (Math.tan(k) * ((k * t_m) / l)));
	} else {
		tmp = 2.0 / (((k / l) * ((t_2 * Math.tan(k)) * t_m)) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sin(k) / l
	tmp = 0
	if t_m <= 18000.0:
		tmp = 2.0 / ((t_2 * k) * (math.tan(k) * ((k * t_m) / l)))
	else:
		tmp = 2.0 / (((k / l) * ((t_2 * math.tan(k)) * t_m)) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 18000.0)
		tmp = Float64(2.0 / Float64(Float64(t_2 * k) * Float64(tan(k) * Float64(Float64(k * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(Float64(t_2 * tan(k)) * t_m)) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) / l;
	tmp = 0.0;
	if (t_m <= 18000.0)
		tmp = 2.0 / ((t_2 * k) * (tan(k) * ((k * t_m) / l)));
	else
		tmp = 2.0 / (((k / l) * ((t_2 * tan(k)) * t_m)) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 18000.0], N[(2.0 / N[(N[(t$95$2 * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 18000

   1. Initial program 39.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 84.6%

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

   9. Applied rewrites 95.4%

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

   11. Applied rewrites 97.8%

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

   if 18000 < t

   1. Initial program 25.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 72.1%

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

   9. Applied rewrites 90.0%

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

   11. Applied rewrites 99.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 18000:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \left(\tan k \cdot \frac{k \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot t\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot k}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \tan k}{\ell \cdot \ell} \cdot k\right) \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.4e-6)
    (/ 2.0 (* (* (/ (* k k) l) (/ (* k t_m) l)) k))
    (if (<= k 7.5e+113)
      (*
       (/
        (* (* (cos k) 2.0) l)
        (* (- 0.5 (* (cos (+ k k)) 0.5)) (* (* k k) t_m)))
       l)
      (/ 2.0 (* (* (/ (* (sin k) (tan k)) (* l l)) k) (* k t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e-6) {
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k);
	} else if (k <= 7.5e+113) {
		tmp = (((cos(k) * 2.0) * l) / ((0.5 - (cos((k + k)) * 0.5)) * ((k * k) * t_m))) * l;
	} else {
		tmp = 2.0 / ((((sin(k) * tan(k)) / (l * l)) * k) * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.4d-6) then
        tmp = 2.0d0 / ((((k * k) / l) * ((k * t_m) / l)) * k)
    else if (k <= 7.5d+113) then
        tmp = (((cos(k) * 2.0d0) * l) / ((0.5d0 - (cos((k + k)) * 0.5d0)) * ((k * k) * t_m))) * l
    else
        tmp = 2.0d0 / ((((sin(k) * tan(k)) / (l * l)) * k) * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e-6) {
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k);
	} else if (k <= 7.5e+113) {
		tmp = (((Math.cos(k) * 2.0) * l) / ((0.5 - (Math.cos((k + k)) * 0.5)) * ((k * k) * t_m))) * l;
	} else {
		tmp = 2.0 / ((((Math.sin(k) * Math.tan(k)) / (l * l)) * k) * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4.4e-6:
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k)
	elif k <= 7.5e+113:
		tmp = (((math.cos(k) * 2.0) * l) / ((0.5 - (math.cos((k + k)) * 0.5)) * ((k * k) * t_m))) * l
	else:
		tmp = 2.0 / ((((math.sin(k) * math.tan(k)) / (l * l)) * k) * (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.4e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * t_m) / l)) * k));
	elseif (k <= 7.5e+113)
		tmp = Float64(Float64(Float64(Float64(cos(k) * 2.0) * l) / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * Float64(Float64(k * k) * t_m))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * tan(k)) / Float64(l * l)) * k) * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4.4e-6)
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k);
	elseif (k <= 7.5e+113)
		tmp = (((cos(k) * 2.0) * l) / ((0.5 - (cos((k + k)) * 0.5)) * ((k * k) * t_m))) * l;
	else
		tmp = 2.0 / ((((sin(k) * tan(k)) / (l * l)) * k) * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.4e-6], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.5e+113], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.4000000000000002e-6

   1. Initial program 39.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 76.5%

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

   if 4.4000000000000002e-6 < k < 7.5000000000000001e113

   1. Initial program 12.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 94.9%

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

   if 7.5000000000000001e113 < k

   1. Initial program 32.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 89.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot k}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(\cos k \cdot 2\right) \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \tan k}{\ell \cdot \ell} \cdot k\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k \cdot k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot t\_2\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(k \cdot t\_m\right)\right) \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (* k k) l)))
   (*
    t_s
    (if (<= (* l l) 0.0)
      (/ 2.0 (* (* t_2 t_2) t_m))
      (/ 2.0 (* (* (* (/ (tan k) (* l l)) (sin k)) (* k t_m)) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) / l;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / ((t_2 * t_2) * t_m);
	} else {
		tmp = 2.0 / ((((tan(k) / (l * l)) * sin(k)) * (k * t_m)) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k * k) / l
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((t_2 * t_2) * t_m)
    else
        tmp = 2.0d0 / ((((tan(k) / (l * l)) * sin(k)) * (k * t_m)) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) / l;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / ((t_2 * t_2) * t_m);
	} else {
		tmp = 2.0 / ((((Math.tan(k) / (l * l)) * Math.sin(k)) * (k * t_m)) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k * k) / l
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / ((t_2 * t_2) * t_m)
	else:
		tmp = 2.0 / ((((math.tan(k) / (l * l)) * math.sin(k)) * (k * t_m)) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * k) / l)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(t_2 * t_2) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) / Float64(l * l)) * sin(k)) * Float64(k * t_m)) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k * k) / l;
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((t_2 * t_2) * t_m);
	else
		tmp = 2.0 / ((((tan(k) / (l * l)) * sin(k)) * (k * t_m)) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

   9. Applied rewrites 86.3%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 43.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 86.5%

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

   9. Applied rewrites 87.0%

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

4. Final simplification 86.8%

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

Alternative 4: 79.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, \mathsf{fma}\left(0.08611111111111111 \cdot k, k, 0.16666666666666666\right), \frac{1}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \tan k}{\ell \cdot \ell} \cdot k\right) \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4e-20)
    (/
     2.0
     (*
      (*
       (*
        (fma
         (/ (* k k) l)
         (fma (* 0.08611111111111111 k) k 0.16666666666666666)
         (/ 1.0 l))
        (* k k))
       (/ (* k t_m) l))
      k))
    (/ 2.0 (* (* (/ (* (sin k) (tan k)) (* l l)) k) (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4e-20) {
		tmp = 2.0 / (((fma(((k * k) / l), fma((0.08611111111111111 * k), k, 0.16666666666666666), (1.0 / l)) * (k * k)) * ((k * t_m) / l)) * k);
	} else {
		tmp = 2.0 / ((((sin(k) * tan(k)) / (l * l)) * k) * (k * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4e-20)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(k * k) / l), fma(Float64(0.08611111111111111 * k), k, 0.16666666666666666), Float64(1.0 / l)) * Float64(k * k)) * Float64(Float64(k * t_m) / l)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * tan(k)) / Float64(l * l)) * k) * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4e-20], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(0.08611111111111111 * k), $MachinePrecision] * k + 0.16666666666666666), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.99999999999999978e-20

   1. Initial program 39.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 76.3%

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

   9. Applied rewrites 93.0%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{k \cdot \left(\frac{t \cdot k}{\ell} \cdot \left({k}^{2} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{6} \cdot \frac{1}{\ell}\right) + \frac{1}{\ell}\right)}\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.1%

       \[\leadsto \frac{2}{k \cdot \left(\frac{t \cdot k}{\ell} \cdot \left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, \mathsf{fma}\left(0.08611111111111111 \cdot k, k, 0.16666666666666666\right), \frac{1}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

   if 3.99999999999999978e-20 < k

   1. Initial program 25.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 85.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, \mathsf{fma}\left(0.08611111111111111 \cdot k, k, 0.16666666666666666\right), \frac{1}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{k \cdot t}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \tan k}{\ell \cdot \ell} \cdot k\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 72.3

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 78.1%

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

9. Applied rewrites 92.8%

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

11. Applied rewrites 93.9%

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

12. Final simplification 93.9%

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

Alternative 6: 92.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 72.3

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 78.1%

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

9. Applied rewrites 93.2%

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

10. Final simplification 93.2%

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

Alternative 7: 73.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+260}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{2 - k \cdot k}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3.8e+260)
    (/ 2.0 (* (* (/ (* k k) l) (/ (* k t_m) l)) k))
    (*
     (/ (* l l) (* (- 0.5 (* (cos (+ k k)) 0.5)) (* k t_m)))
     (/ (- 2.0 (* k k)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.8e+260) {
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k);
	} else {
		tmp = ((l * l) / ((0.5 - (cos((k + k)) * 0.5)) * (k * t_m))) * ((2.0 - (k * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 3.8d+260) then
        tmp = 2.0d0 / ((((k * k) / l) * ((k * t_m) / l)) * k)
    else
        tmp = ((l * l) / ((0.5d0 - (cos((k + k)) * 0.5d0)) * (k * t_m))) * ((2.0d0 - (k * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 3.8e+260) {
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k);
	} else {
		tmp = ((l * l) / ((0.5 - (Math.cos((k + k)) * 0.5)) * (k * t_m))) * ((2.0 - (k * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 3.8e+260:
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k)
	else:
		tmp = ((l * l) / ((0.5 - (math.cos((k + k)) * 0.5)) * (k * t_m))) * ((2.0 - (k * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 3.8e+260)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * t_m) / l)) * k));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * Float64(k * t_m))) * Float64(Float64(2.0 - Float64(k * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 3.8e+260)
		tmp = 2.0 / ((((k * k) / l) * ((k * t_m) / l)) * k);
	else
		tmp = ((l * l) / ((0.5 - (cos((k + k)) * 0.5)) * (k * t_m))) * ((2.0 - (k * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 3.8e+260], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 - N[(k * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.7999999999999997e260

   1. Initial program 36.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 72.6%

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

   if 3.7999999999999997e260 < l

   1. Initial program 20.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 60.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+260}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\right)} \cdot \frac{2 - k \cdot k}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 72.3

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 78.1%

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

9. Applied rewrites 92.8%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 71.6%

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

13. Final simplification 71.6%

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

Alternative 9: 73.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 71.4%

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

8. Final simplification 71.4%

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

Alternative 10: 71.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

8. Taylor expanded in l around 0

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 70.5%

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

13. Final simplification 70.5%

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

Alternative 11: 70.2% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

9. Applied rewrites 69.4%

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

10. Final simplification 69.4%

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

Alternative 12: 69.5% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 61.0

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 69.3%

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

8. Final simplification 69.3%

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

Alternative 13: 68.5% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 72.3

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 84.2%

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

9. Applied rewrites 93.4%

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

10. Taylor expanded in k around 0

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

    1. associate-*r/ N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. count-2 N/A

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

    7. lower-+.f64 N/A

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

    8. lower-/.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. metadata-eval N/A

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

    12. pow-sqr N/A

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

    13. lower-*.f64 N/A

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

    14. unpow2 N/A

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

    15. lower-*.f64 N/A

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

    16. unpow2 N/A

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

    17. lower-*.f64 68.1

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

12. Applied rewrites 68.1%

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

13. Final simplification 68.1%

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

Alternative 14: 30.2% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-1}{3} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   4. *-commutative N/A

      \[\leadsto \frac{{k}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

5. Applied rewrites 44.7%

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

6. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
7. Step-by-step derivation

8. Applied rewrites 32.0%

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

9. Final simplification 32.0%

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

Reproduce

herbie shell --seed 2024235 
(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))))