VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.7% → 87.1%

Time: 27.1s

Alternatives: 11

Speedup: 7.1×

• could not determine a ground truth

  1. F = 6.794205321283442e+179
  2. l = 2603136613992828.5

Specification

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{\tan \left(\pi \cdot l\_m\right)}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\sin \left(\pi \cdot l\_m\right)}{{l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(-0.001388888888888889 \cdot \left({l\_m}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right) + 1}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 10000000000.0)
    (+ (* PI l_m) (* (/ (tan (* PI l_m)) F) (/ -1.0 F)))
    (-
     (* PI l_m)
     (/
      (/
       (sin (* PI l_m))
       (+
        (*
         (pow l_m 2.0)
         (+
          (* -0.5 (pow PI 2.0))
          (*
           (pow l_m 2.0)
           (+
            (* -0.001388888888888889 (* (pow l_m 2.0) (pow PI 6.0)))
            (* 0.041666666666666664 (pow PI 4.0))))))
        1.0))
      (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 10000000000.0) {
		tmp = (((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) / F) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) - ((sin((((double) M_PI) * l_m)) / ((pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((-0.001388888888888889 * (pow(l_m, 2.0) * pow(((double) M_PI), 6.0))) + (0.041666666666666664 * pow(((double) M_PI), 4.0)))))) + 1.0)) / (F * F));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 10000000000.0) {
		tmp = (Math.PI * l_m) + ((Math.tan((Math.PI * l_m)) / F) * (-1.0 / F));
	} else {
		tmp = (Math.PI * l_m) - ((Math.sin((Math.PI * l_m)) / ((Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 6.0))) + (0.041666666666666664 * Math.pow(Math.PI, 4.0)))))) + 1.0)) / (F * F));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 10000000000.0:
		tmp = (math.pi * l_m) + ((math.tan((math.pi * l_m)) / F) * (-1.0 / F))
	else:
		tmp = (math.pi * l_m) - ((math.sin((math.pi * l_m)) / ((math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((-0.001388888888888889 * (math.pow(l_m, 2.0) * math.pow(math.pi, 6.0))) + (0.041666666666666664 * math.pow(math.pi, 4.0)))))) + 1.0)) / (F * F))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 10000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(pi * l_m)) / F) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(sin(Float64(pi * l_m)) / Float64(Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * (pi ^ 6.0))) + Float64(0.041666666666666664 * (pi ^ 4.0)))))) + 1.0)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 10000000000.0)
		tmp = (pi * l_m) + ((tan((pi * l_m)) / F) * (-1.0 / F));
	else
		tmp = (pi * l_m) - ((sin((pi * l_m)) / (((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + ((l_m ^ 2.0) * ((-0.001388888888888889 * ((l_m ^ 2.0) * (pi ^ 6.0))) + (0.041666666666666664 * (pi ^ 4.0)))))) + 1.0)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 10000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e10

   1. Initial program 80.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 80.0%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 80.0%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 80.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 80.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 80.6%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 89.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

      2. div-inv 89.5%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   6. Applied egg-rr 89.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   if 1e10 < (*.f64 (PI.f64) l)

   1. Initial program 61.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 61.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 61.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 61.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 75.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{{\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right) + 1}}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{\tan \left(\pi \cdot l\_m\right)}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\sin \left(\pi \cdot l\_m\right)}{{l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({l\_m}^{2} \cdot {\pi}^{4}\right)\right) + 1}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 10000000000.0)
    (+ (* PI l_m) (* (/ (tan (* PI l_m)) F) (/ -1.0 F)))
    (-
     (* PI l_m)
     (/
      (/
       (sin (* PI l_m))
       (+
        (*
         (pow l_m 2.0)
         (+
          (* -0.5 (pow PI 2.0))
          (* 0.041666666666666664 (* (pow l_m 2.0) (pow PI 4.0)))))
        1.0))
      (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 10000000000.0) {
		tmp = (((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) / F) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) - ((sin((((double) M_PI) * l_m)) / ((pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (0.041666666666666664 * (pow(l_m, 2.0) * pow(((double) M_PI), 4.0))))) + 1.0)) / (F * F));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 10000000000.0) {
		tmp = (Math.PI * l_m) + ((Math.tan((Math.PI * l_m)) / F) * (-1.0 / F));
	} else {
		tmp = (Math.PI * l_m) - ((Math.sin((Math.PI * l_m)) / ((Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (0.041666666666666664 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 4.0))))) + 1.0)) / (F * F));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 10000000000.0:
		tmp = (math.pi * l_m) + ((math.tan((math.pi * l_m)) / F) * (-1.0 / F))
	else:
		tmp = (math.pi * l_m) - ((math.sin((math.pi * l_m)) / ((math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (0.041666666666666664 * (math.pow(l_m, 2.0) * math.pow(math.pi, 4.0))))) + 1.0)) / (F * F))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 10000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(pi * l_m)) / F) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(sin(Float64(pi * l_m)) / Float64(Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64(0.041666666666666664 * Float64((l_m ^ 2.0) * (pi ^ 4.0))))) + 1.0)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 10000000000.0)
		tmp = (pi * l_m) + ((tan((pi * l_m)) / F) * (-1.0 / F));
	else
		tmp = (pi * l_m) - ((sin((pi * l_m)) / (((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + (0.041666666666666664 * ((l_m ^ 2.0) * (pi ^ 4.0))))) + 1.0)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 10000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e10

   1. Initial program 80.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 80.0%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 80.0%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 80.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 80.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 80.6%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 89.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

      2. div-inv 89.5%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   6. Applied egg-rr 89.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   if 1e10 < (*.f64 (PI.f64) l)

   1. Initial program 61.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 61.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 61.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 61.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 71.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right)}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{{\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right) + 1}}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{\tan \left(\pi \cdot l\_m\right)}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\sin \left(\pi \cdot l\_m\right)}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot l\_m\right)}^{2}, 1\right)}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 10000000000.0)
    (+ (* PI l_m) (* (/ (tan (* PI l_m)) F) (/ -1.0 F)))
    (-
     (* PI l_m)
     (/ (/ (sin (* PI l_m)) (fma -0.5 (pow (* PI l_m) 2.0) 1.0)) (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 10000000000.0) {
		tmp = (((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) / F) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) - ((sin((((double) M_PI) * l_m)) / fma(-0.5, pow((((double) M_PI) * l_m), 2.0), 1.0)) / (F * F));
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 10000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(pi * l_m)) / F) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(sin(Float64(pi * l_m)) / fma(-0.5, (Float64(pi * l_m) ^ 2.0), 1.0)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 10000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e10

   1. Initial program 80.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 80.0%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 80.0%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 80.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 80.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 80.6%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 89.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

      2. div-inv 89.5%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   6. Applied egg-rr 89.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   if 1e10 < (*.f64 (PI.f64) l)

   1. Initial program 61.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 61.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 61.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 61.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 61.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 61.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 61.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 71.8%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{1 + -0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}}}{F \cdot F} \]
   8. Step-by-step derivation

      1. +-commutative 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{-0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right) + 1}}}{F \cdot F} \]

      2. fma-define 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}}{F \cdot F} \]

      3. *-commutative 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}}{F \cdot F} \]

      4. unpow2 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\pi}^{2} \cdot \color{blue}{\left(\ell \cdot \ell\right)}, 1\right)}}{F \cdot F} \]

      5. unpow2 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(\ell \cdot \ell\right), 1\right)}}{F \cdot F} \]

      6. swap-sqr 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)}, 1\right)}}{F \cdot F} \]

      7. unpow2 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}}{F \cdot F} \]

      8. *-commutative 71.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}}{F \cdot F} \]

   9. Simplified 71.8%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{\tan \left(l\_m \cdot \sqrt[3]{{\pi}^{3}}\right)}{F} \cdot \frac{-1}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (* (/ (tan (* l_m (cbrt (pow PI 3.0)))) F) (/ -1.0 F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((tan((l_m * cbrt(pow(((double) M_PI), 3.0)))) / F) * (-1.0 / F)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + ((Math.tan((l_m * Math.cbrt(Math.pow(Math.PI, 3.0)))) / F) * (-1.0 / F)));
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(l_m * cbrt((pi ^ 3.0)))) / F) * Float64(-1.0 / F))))
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(l$95$m * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 76.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 76.0%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 76.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. *-rgt-identity 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

   5. sqr-neg 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

3. Simplified 76.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 83.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   2. div-inv 83.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

6. Applied egg-rr 83.3%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]
7. Step-by-step derivation

   1. add-cbrt-cube 83.4%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}} \cdot \ell\right)}{F} \cdot \frac{1}{F} \]

   2. pow3 83.4%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\color{blue}{{\pi}^{3}}} \cdot \ell\right)}{F} \cdot \frac{1}{F} \]

8. Applied egg-rr 83.4%

   \[\leadsto \pi \cdot \ell - \frac{\tan \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \ell\right)}{F} \cdot \frac{1}{F} \]

9. Final simplification 83.4%

   \[\leadsto \pi \cdot \ell + \frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F} \cdot \frac{-1}{F} \]
10. Add Preprocessing

Alternative 5: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\pi \cdot l\_m + \left(\pi \cdot \frac{l\_m}{F}\right) \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-47)
    (+ (* PI l_m) (* (* PI (/ l_m F)) (/ -1.0 F)))
    (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e-47) {
		tmp = (((double) M_PI) * l_m) + ((((double) M_PI) * (l_m / F)) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 2e-47) {
		tmp = (Math.PI * l_m) + ((Math.PI * (l_m / F)) * (-1.0 / F));
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 2e-47:
		tmp = (math.pi * l_m) + ((math.pi * (l_m / F)) * (-1.0 / F))
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) / (F * F))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e-47)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(pi * Float64(l_m / F)) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 2e-47)
		tmp = (pi * l_m) + ((pi * (l_m / F)) * (-1.0 / F));
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-47], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.9999999999999999e-47

   1. Initial program 79.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 79.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 79.6%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 79.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 79.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 79.8%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 79.8%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 74.6%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. *-un-lft-identity 74.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{1 \cdot \frac{\pi \cdot \ell}{F \cdot F}} \]

      3. associate-*r/ 74.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      4. times-frac 84.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \frac{\pi \cdot \ell}{F}} \]

      5. associate-/l* 84.1%

         \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\left(\pi \cdot \frac{\ell}{F}\right)} \]

   7. Applied egg-rr 84.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\pi \cdot \frac{\ell}{F}\right)} \]

   if 1.9999999999999999e-47 < (*.f64 (PI.f64) l)

   1. Initial program 66.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 66.4%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 67.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 67.7%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 67.7%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 67.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \frac{\ell}{F}\right) \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 1.0× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (* (/ (tan (* PI l_m)) F) (/ -1.0 F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) / F) * (-1.0 / F)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + ((Math.tan((Math.PI * l_m)) / F) * (-1.0 / F)));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + ((math.tan((math.pi * l_m)) / F) * (-1.0 / F)))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(pi * l_m)) / F) * Float64(-1.0 / F))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + ((tan((pi * l_m)) / F) * (-1.0 / F)));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 76.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 76.0%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 76.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. *-rgt-identity 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

   5. sqr-neg 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

3. Simplified 76.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 83.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   2. div-inv 83.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

6. Applied egg-rr 83.3%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

7. Final simplification 83.3%

   \[\leadsto \pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F} \]
8. Add Preprocessing

Alternative 7: 81.7% accurate, 1.0× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F)))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((tan((pi * l_m)) / F) / F));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 76.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   2. *-un-lft-identity 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. associate-/r* 83.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

4. Applied egg-rr 83.3%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

5. Final simplification 83.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F} \]
6. Add Preprocessing

Alternative 8: 75.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\pi \cdot l\_m + \left(\pi \cdot \frac{l\_m}{F}\right) \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 0.5)
    (+ (* PI l_m) (* (* PI (/ l_m F)) (/ -1.0 F)))
    (+ (* PI l_m) (/ (* l_m (/ PI F)) F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = (((double) M_PI) * l_m) + ((((double) M_PI) * (l_m / F)) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) + ((l_m * (((double) M_PI) / F)) / F);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = (Math.PI * l_m) + ((Math.PI * (l_m / F)) * (-1.0 / F));
	} else {
		tmp = (Math.PI * l_m) + ((l_m * (Math.PI / F)) / F);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 0.5:
		tmp = (math.pi * l_m) + ((math.pi * (l_m / F)) * (-1.0 / F))
	else:
		tmp = (math.pi * l_m) + ((l_m * (math.pi / F)) / F)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 0.5)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(pi * Float64(l_m / F)) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(l_m * Float64(pi / F)) / F));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 0.5)
		tmp = (pi * l_m) + ((pi * (l_m / F)) * (-1.0 / F));
	else
		tmp = (pi * l_m) + ((l_m * (pi / F)) / F);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 0.5], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 80.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 80.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 80.1%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 80.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 80.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 80.8%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 80.8%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 75.8%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. *-un-lft-identity 75.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{1 \cdot \frac{\pi \cdot \ell}{F \cdot F}} \]

      3. associate-*r/ 75.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      4. times-frac 84.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \frac{\pi \cdot \ell}{F}} \]

      5. associate-/l* 84.7%

         \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\left(\pi \cdot \frac{\ell}{F}\right)} \]

   7. Applied egg-rr 84.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \left(\pi \cdot \frac{\ell}{F}\right)} \]

   if 0.5 < l

   1. Initial program 62.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 62.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      2. *-un-lft-identity 62.4%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      3. associate-/r* 62.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

      4. frac-2neg 62.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}} \]

      5. add-sqr-sqrt 38.2%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{-F} \cdot \sqrt{-F}}} \]

      6. sqrt-unprod 61.7%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{\left(-F\right) \cdot \left(-F\right)}}} \]

      7. sqr-neg 61.7%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{\color{blue}{F \cdot F}}} \]

      8. sqrt-prod 23.5%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}} \]

      9. add-sqr-sqrt 59.6%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F}} \]

   4. Applied egg-rr 59.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   5. Taylor expanded in l around 0 54.7%

      \[\leadsto \pi \cdot \ell - \frac{-\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]
   6. Step-by-step derivation

      1. associate-/l* 54.7%

         \[\leadsto \pi \cdot \ell - \frac{-\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]

   7. Simplified 54.7%

      \[\leadsto \pi \cdot \ell - \frac{-\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \frac{\ell}{F}\right) \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F \cdot \frac{F}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 0.5)
    (- (* PI l_m) (/ PI (* F (/ F l_m))))
    (+ (* PI l_m) (/ (* l_m (/ PI F)) F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = (((double) M_PI) * l_m) - (((double) M_PI) / (F * (F / l_m)));
	} else {
		tmp = (((double) M_PI) * l_m) + ((l_m * (((double) M_PI) / F)) / F);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = (Math.PI * l_m) - (Math.PI / (F * (F / l_m)));
	} else {
		tmp = (Math.PI * l_m) + ((l_m * (Math.PI / F)) / F);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 0.5:
		tmp = (math.pi * l_m) - (math.pi / (F * (F / l_m)))
	else:
		tmp = (math.pi * l_m) + ((l_m * (math.pi / F)) / F)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 0.5)
		tmp = Float64(Float64(pi * l_m) - Float64(pi / Float64(F * Float64(F / l_m))));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(l_m * Float64(pi / F)) / F));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 0.5)
		tmp = (pi * l_m) - (pi / (F * (F / l_m)));
	else
		tmp = (pi * l_m) + ((l_m * (pi / F)) / F);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 0.5], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(Pi / N[(F * N[(F / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 80.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 80.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 80.1%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 80.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. *-rgt-identity 80.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

      5. sqr-neg 80.8%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Simplified 80.8%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 75.8%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. times-frac 84.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   7. Applied egg-rr 84.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
   8. Step-by-step derivation

      1. *-commutative 84.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]

      2. clear-num 84.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{\frac{F}{\ell}}} \cdot \frac{\pi}{F} \]

      3. frac-times 84.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \pi}{\frac{F}{\ell} \cdot F}} \]

      4. *-un-lft-identity 84.7%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi}}{\frac{F}{\ell} \cdot F} \]

   9. Applied egg-rr 84.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{\frac{F}{\ell} \cdot F}} \]

   if 0.5 < l

   1. Initial program 62.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 62.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      2. *-un-lft-identity 62.4%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      3. associate-/r* 62.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

      4. frac-2neg 62.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}} \]

      5. add-sqr-sqrt 38.2%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{-F} \cdot \sqrt{-F}}} \]

      6. sqrt-unprod 61.7%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{\left(-F\right) \cdot \left(-F\right)}}} \]

      7. sqr-neg 61.7%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{\color{blue}{F \cdot F}}} \]

      8. sqrt-prod 23.5%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}} \]

      9. add-sqr-sqrt 59.6%

         \[\leadsto \pi \cdot \ell - \frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F}} \]

   4. Applied egg-rr 59.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   5. Taylor expanded in l around 0 54.7%

      \[\leadsto \pi \cdot \ell - \frac{-\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]
   6. Step-by-step derivation

      1. associate-/l* 54.7%

         \[\leadsto \pi \cdot \ell - \frac{-\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]

   7. Simplified 54.7%

      \[\leadsto \pi \cdot \ell - \frac{-\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 10.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ l_m F) (/ PI F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / F)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((l_m / F) * (Math.PI / F)));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((l_m / F) * (math.pi / F)))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / F))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((l_m / F) * (pi / F)));
end

Wolfram

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

Derivation

1. Initial program 76.0%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 76.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 76.0%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 76.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. *-rgt-identity 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

   5. sqr-neg 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

3. Simplified 76.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 69.8%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 69.8%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 76.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 76.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

8. Final simplification 76.6%

   \[\leadsto \pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F} \]
9. Add Preprocessing

Alternative 11: 74.1% accurate, 10.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ PI (* F (/ F l_m))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - (((double) M_PI) / (F * (F / l_m))));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - (Math.PI / (F * (F / l_m))));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - (math.pi / (F * (F / l_m))))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(pi / Float64(F * Float64(F / l_m)))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - (pi / (F * (F / l_m))));
end

Wolfram

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

Derivation

1. Initial program 76.0%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 76.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 76.0%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 76.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. *-rgt-identity 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{\left(-F\right) \cdot \left(-F\right)} \]

   5. sqr-neg 76.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

3. Simplified 76.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 69.8%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 69.8%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 76.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 76.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
8. Step-by-step derivation

   1. *-commutative 76.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]

   2. clear-num 76.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{\frac{F}{\ell}}} \cdot \frac{\pi}{F} \]

   3. frac-times 76.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \pi}{\frac{F}{\ell} \cdot F}} \]

   4. *-un-lft-identity 76.6%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi}}{\frac{F}{\ell} \cdot F} \]

9. Applied egg-rr 76.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{\frac{F}{\ell} \cdot F}} \]

10. Final simplification 76.6%

    \[\leadsto \pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))