Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.0% → 90.6%

Time: 16.4s

Alternatives: 10

Speedup: 17.1×

• 38.8% 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: 35.0% 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: 90.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan k \cdot \sin k\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k \cdot k}{\ell} \cdot t\_1\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+190}:\\ \;\;\;\;\left(\ell \cdot \left(\ell + \ell\right)\right) \cdot \frac{\cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (tan k) (sin k))))
   (if (<= (* l l) 5e-211)
     (/ 2.0 (* (* t (* (/ (* k k) l) t_1)) (/ 1.0 l)))
     (if (<= (* l l) 5e+190)
       (* (* l (+ l l)) (/ (cos k) (* k (* k (* t (pow (sin k) 2.0))))))
       (/ 2.0 (* (/ t l) (* (/ k l) (* k t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = tan(k) * sin(k);
	double tmp;
	if ((l * l) <= 5e-211) {
		tmp = 2.0 / ((t * (((k * k) / l) * t_1)) * (1.0 / l));
	} else if ((l * l) <= 5e+190) {
		tmp = (l * (l + l)) * (cos(k) / (k * (k * (t * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 / ((t / l) * ((k / l) * (k * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = tan(k) * sin(k)
    if ((l * l) <= 5d-211) then
        tmp = 2.0d0 / ((t * (((k * k) / l) * t_1)) * (1.0d0 / l))
    else if ((l * l) <= 5d+190) then
        tmp = (l * (l + l)) * (cos(k) / (k * (k * (t * (sin(k) ** 2.0d0)))))
    else
        tmp = 2.0d0 / ((t / l) * ((k / l) * (k * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.tan(k) * Math.sin(k);
	double tmp;
	if ((l * l) <= 5e-211) {
		tmp = 2.0 / ((t * (((k * k) / l) * t_1)) * (1.0 / l));
	} else if ((l * l) <= 5e+190) {
		tmp = (l * (l + l)) * (Math.cos(k) / (k * (k * (t * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 / ((t / l) * ((k / l) * (k * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.tan(k) * math.sin(k)
	tmp = 0
	if (l * l) <= 5e-211:
		tmp = 2.0 / ((t * (((k * k) / l) * t_1)) * (1.0 / l))
	elif (l * l) <= 5e+190:
		tmp = (l * (l + l)) * (math.cos(k) / (k * (k * (t * math.pow(math.sin(k), 2.0)))))
	else:
		tmp = 2.0 / ((t / l) * ((k / l) * (k * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(tan(k) * sin(k))
	tmp = 0.0
	if (Float64(l * l) <= 5e-211)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(Float64(k * k) / l) * t_1)) * Float64(1.0 / l)));
	elseif (Float64(l * l) <= 5e+190)
		tmp = Float64(Float64(l * Float64(l + l)) * Float64(cos(k) / Float64(k * Float64(k * Float64(t * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(k / l) * Float64(k * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = tan(k) * sin(k);
	tmp = 0.0;
	if ((l * l) <= 5e-211)
		tmp = 2.0 / ((t * (((k * k) / l) * t_1)) * (1.0 / l));
	elseif ((l * l) <= 5e+190)
		tmp = (l * (l + l)) * (cos(k) / (k * (k * (t * (sin(k) ^ 2.0)))));
	else
		tmp = 2.0 / ((t / l) * ((k / l) * (k * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-211], N[(2.0 / N[(N[(t * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+190], N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * N[(k * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-211

   1. Initial program 31.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 42.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 70.6

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

   7. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 85.7%

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

   if 5.0000000000000002e-211 < (*.f64 l l) < 5.00000000000000036e190

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. count-2 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-sin.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 5.00000000000000036e190 < (*.f64 l l)

   1. Initial program 35.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 82.0

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

   7. Applied rewrites 82.0%

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

   9. Applied rewrites 93.5%

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

Alternative 2: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-74)
   (* (/ (+ l l) (* t (* k k))) (/ l (* k k)))
   (* l (/ 2.0 (* t (* (* (tan k) (sin k)) (* k (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-74) {
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
	} else {
		tmp = l * (2.0 / (t * ((tan(k) * sin(k)) * (k * (k / l)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-74) then
        tmp = ((l + l) / (t * (k * k))) * (l / (k * k))
    else
        tmp = l * (2.0d0 / (t * ((tan(k) * sin(k)) * (k * (k / l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-74) {
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
	} else {
		tmp = l * (2.0 / (t * ((Math.tan(k) * Math.sin(k)) * (k * (k / l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.8e-74:
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k))
	else:
		tmp = l * (2.0 / (t * ((math.tan(k) * math.sin(k)) * (k * (k / l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-74)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k * k))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(l * Float64(2.0 / Float64(t * Float64(Float64(tan(k) * sin(k)) * Float64(k * Float64(k / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.8e-74)
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
	else
		tmp = l * (2.0 / (t * ((tan(k) * sin(k)) * (k * (k / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.8e-74], N[(N[(N[(l + l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(t * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e-74

   1. Initial program 39.6%

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

   3. Taylor expanded in 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 78.9%

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

   if 5.8e-74 < k

   1. Initial program 28.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 33.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 85.1

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

   7. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 84.2%

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

   11. Applied rewrites 92.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (+ l l) (* t (* (sin k) (* (/ (* k k) l) (tan k))))))

C

double code(double t, double l, double k) {
	return (l + l) / (t * (sin(k) * (((k * k) / l) * tan(k))));
}

Fortran

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

Java

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

Python

def code(t, l, k):
	return (l + l) / (t * (math.sin(k) * (((k * k) / l) * math.tan(k))))

Julia

function code(t, l, k)
	return Float64(Float64(l + l) / Float64(t * Float64(sin(k) * Float64(Float64(Float64(k * k) / l) * tan(k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) / (t * (sin(k) * (((k * k) / l) * tan(k))));
end

Wolfram

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

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 35.4%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 78.2

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

7. Applied rewrites 78.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

9. Applied rewrites 85.2%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

    4. count-2 N/A

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

    5. lift-+.f64 N/A

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

    6. lower-/.f64 85.2

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

11. Applied rewrites 85.2%

    \[\leadsto \color{blue}{\frac{\ell + \ell}{t \cdot \left(\sin k \cdot \left(\frac{k \cdot k}{\ell} \cdot \tan k\right)\right)}} \]
12. Add Preprocessing

Alternative 4: 73.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 10000000000000:\\ \;\;\;\;\frac{\ell + \ell}{k \cdot k} \cdot \frac{\frac{\ell}{t}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 10000000000000.0)
   (* (/ (+ l l) (* k k)) (/ (/ l t) (* k k)))
   (* (/ (+ l l) (* t (* k k))) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 10000000000000.0) {
		tmp = ((l + l) / (k * k)) * ((l / t) / (k * k));
	} else {
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 10000000000000.0d0) then
        tmp = ((l + l) / (k * k)) * ((l / t) / (k * k))
    else
        tmp = ((l + l) / (t * (k * k))) * (l / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 10000000000000.0) {
		tmp = ((l + l) / (k * k)) * ((l / t) / (k * k));
	} else {
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 10000000000000.0:
		tmp = ((l + l) / (k * k)) * ((l / t) / (k * k))
	else:
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 10000000000000.0)
		tmp = Float64(Float64(Float64(l + l) / Float64(k * k)) * Float64(Float64(l / t) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k * k))) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 10000000000000.0)
		tmp = ((l + l) / (k * k)) * ((l / t) / (k * k));
	else
		tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 10000000000000.0], N[(N[(N[(l + l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1e13

   1. Initial program 37.4%

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

   3. Taylor expanded in 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 42.0%

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

   9. Applied rewrites 38.0%

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

   11. Applied rewrites 70.9%

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

   if 1e13 < t

   1. Initial program 31.4%

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

   3. Taylor expanded in 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 81.3%

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

Alternative 5: 73.2% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell + \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (+ l l) (* t (* k k))) (/ l (* k k))))

C

double code(double t, double l, double k) {
	return ((l + l) / (t * (k * k))) * (l / (k * k));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l + l) / (t * (k * k))) * (l / (k * k))
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) / (t * (k * k))) * (l / (k * k));
}

Python

def code(t, l, k):
	return ((l + l) / (t * (k * k))) * (l / (k * k))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) / Float64(t * Float64(k * k))) * Float64(l / Float64(k * k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) / (t * (k * k))) * (l / (k * k));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in 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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 73.6%

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

Alternative 6: 71.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell + \ell}{t \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot k\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (+ l l) (* t k)) (/ l (* k (* k k)))))

C

double code(double t, double l, double k) {
	return ((l + l) / (t * k)) * (l / (k * (k * k)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l + l) / (t * k)) * (l / (k * (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return ((l + l) / (t * k)) * (l / (k * (k * k)));
}

Python

def code(t, l, k):
	return ((l + l) / (t * k)) * (l / (k * (k * k)))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l + l) / Float64(t * k)) * Float64(l / Float64(k * Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l + l) / (t * k)) * (l / (k * (k * k)));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l + l), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in 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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 46.0%

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

9. Applied rewrites 40.9%

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

11. Applied rewrites 71.0%

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

Alternative 7: 69.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \left(\ell + \ell\right) \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (+ l l) (/ l (* k (* t (* k (* k k)))))))

C

double code(double t, double l, double k) {
	return (l + l) * (l / (k * (t * (k * (k * k)))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l + l) * (l / (k * (t * (k * (k * k)))))
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) * (l / (k * (t * (k * (k * k)))));
}

Python

def code(t, l, k):
	return (l + l) * (l / (k * (t * (k * (k * k)))))

Julia

function code(t, l, k)
	return Float64(Float64(l + l) * Float64(l / Float64(k * Float64(t * Float64(k * Float64(k * k))))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) * (l / (k * (t * (k * (k * k)))));
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] * N[(l / N[(k * N[(t * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in 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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 46.0%

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

9. Applied rewrites 40.9%

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

11. Applied rewrites 69.5%

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

12. Final simplification 69.5%

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

Alternative 8: 47.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{\ell + \ell}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (/ (+ l l) (* t (* k (* k k))))))

C

double code(double t, double l, double k) {
	return l * ((l + l) / (t * (k * (k * k))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * ((l + l) / (t * (k * (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return l * ((l + l) / (t * (k * (k * k))));
}

Python

def code(t, l, k):
	return l * ((l + l) / (t * (k * (k * k))))

Julia

function code(t, l, k)
	return Float64(l * Float64(Float64(l + l) / Float64(t * Float64(k * Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * ((l + l) / (t * (k * (k * k))));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(N[(l + l), $MachinePrecision] / N[(t * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in 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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 46.0%

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

9. Applied rewrites 40.9%

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

11. Applied rewrites 45.8%

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

12. Final simplification 45.8%

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

Alternative 9: 42.7% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \frac{\ell + \ell}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (+ l l) (* t (* k (* k k)))))

C

double code(double t, double l, double k) {
	return (l + l) / (t * (k * (k * k)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l + l) / (t * (k * (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return (l + l) / (t * (k * (k * k)));
}

Python

def code(t, l, k):
	return (l + l) / (t * (k * (k * k)))

Julia

function code(t, l, k)
	return Float64(Float64(l + l) / Float64(t * Float64(k * Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l + l) / (t * (k * (k * k)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l + l), $MachinePrecision] / N[(t * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in 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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 46.0%

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

9. Applied rewrites 40.9%

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

11. Applied rewrites 42.9%

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

12. Final simplification 42.9%

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

Alternative 10: 37.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* k (* t (* k k)))))

C

double code(double t, double l, double k) {
	return 2.0 / (k * (t * (k * k)));
}

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 / (k * (t * (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (k * (t * (k * k)));
}

Python

def code(t, l, k):
	return 2.0 / (k * (t * (k * k)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(k * Float64(t * Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (k * (t * (k * k)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(k * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

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

3. Taylor expanded in 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. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 46.0%

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

9. Applied rewrites 40.9%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 39.2%

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

13. Final simplification 39.2%

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

Reproduce

herbie shell --seed 2024223 
(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))))