VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.7% → 99.2%

Time: 17.4s

Alternatives: 13

Speedup: 3.7×

• could not determine a ground truth

  1. F = -2.9387121224535758e+209
  2. l = 2268112272077526.5

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: 76.7% 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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{l\_m}{-F}, \frac{\pi}{F}, \pi \cdot l\_m\right)\\ \mathbf{elif}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 1e-50)
    (fma (/ l_m (- F)) (/ PI F) (* PI l_m))
    (if (<= (* PI l_m) 5e+15)
      (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F))))
      (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI))))))))

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) <= 1e-50) {
		tmp = fma((l_m / -F), (((double) M_PI) / F), (((double) M_PI) * l_m));
	} else if ((((double) M_PI) * l_m) <= 5e+15) {
		tmp = (((double) M_PI) * l_m) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 1e-50)
		tmp = fma(Float64(l_m / Float64(-F)), Float64(pi / F), Float64(pi * l_m));
	elseif (Float64(pi * l_m) <= 5e+15)
		tmp = Float64(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F))));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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], 1e-50], N[(N[(l$95$m / (-F)), $MachinePrecision] * N[(Pi / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e+15], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 1.00000000000000001e-50

   1. Initial program 82.5%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F} \cdot \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(F\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-neg N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\frac{\frac{-1}{F} \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(F\right)}\right)\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{-1}{F} \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(F\right)}}\right)\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\frac{\frac{-1}{F} \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\mathsf{neg}\left(F\right)}}\right)\right) \]

      12. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-1}{F} \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{F}\right)\right)}\right)\right) \]

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 79.7%

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

   if 1.00000000000000001e-50 < (*.f64 (PI.f64) l) < 5e15

   1. Initial program 99.7%

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

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\ell}{-F}, \frac{\pi}{F}, \pi \cdot \ell\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{\pi \cdot l\_m}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -1e-294)
    (/ (* PI l_m) (* F (- F)))
    (* PI l_m))))

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) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = (((double) M_PI) * l_m) / (F * -F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	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) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = (Math.PI * l_m) / (F * -F);
	} else {
		tmp = Math.PI * l_m;
	}
	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) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294:
		tmp = (math.pi * l_m) / (F * -F)
	else:
		tmp = math.pi * l_m
	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(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F)))) <= -1e-294)
		tmp = Float64(Float64(pi * l_m) / Float64(F * Float64(-F)));
	else
		tmp = Float64(pi * l_m);
	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) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294)
		tmp = (pi * l_m) / (F * -F);
	else
		tmp = pi * l_m;
	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[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-294], N[(N[(Pi * l$95$m), $MachinePrecision] / N[(F * (-F)), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.00000000000000002e-294

   1. Initial program 79.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      3. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      6. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{F \cdot F}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{F \cdot F}}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{F \cdot F}}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{F \cdot F}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}{F \cdot F}} \]

   10. Applied rewrites 25.2%

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

   if -1.00000000000000002e-294 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 79.5%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 77.5

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 77.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{\pi \cdot \ell}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;l\_m \cdot \frac{\pi}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -1e-294)
    (* l_m (/ PI (* F (- F))))
    (* PI l_m))))

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) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = l_m * (((double) M_PI) / (F * -F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	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) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)))) <= -1e-294) {
		tmp = l_m * (Math.PI / (F * -F));
	} else {
		tmp = Math.PI * l_m;
	}
	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) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294:
		tmp = l_m * (math.pi / (F * -F))
	else:
		tmp = math.pi * l_m
	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(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F)))) <= -1e-294)
		tmp = Float64(l_m * Float64(pi / Float64(F * Float64(-F))));
	else
		tmp = Float64(pi * l_m);
	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) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -1e-294)
		tmp = l_m * (pi / (F * -F));
	else
		tmp = pi * l_m;
	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[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-294], N[(l$95$m * N[(Pi / N[(F * (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.00000000000000002e-294

   1. Initial program 79.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      3. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \ell \cdot \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      6. lower-*.f64 25.2

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

   8. Applied rewrites 25.2%

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

   if -1.00000000000000002e-294 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 79.5%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 77.5

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 77.5%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\ell \cdot \frac{\pi}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(l\_m, l\_m \cdot \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi\right)}{-F}, \frac{l\_m}{F}, \pi \cdot l\_m\right)\\ \mathbf{elif}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot \left(-F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 1e-7)
    (fma
     (/ (fma l_m (* l_m (* 0.3333333333333333 (* PI (* PI PI)))) PI) (- F))
     (/ l_m F)
     (* PI l_m))
    (if (<= (* PI l_m) 5e+15)
      (fma PI l_m (/ (tan (* PI l_m)) (* F (- F))))
      (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI))))))))

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) <= 1e-7) {
		tmp = fma((fma(l_m, (l_m * (0.3333333333333333 * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), ((double) M_PI)) / -F), (l_m / F), (((double) M_PI) * l_m));
	} else if ((((double) M_PI) * l_m) <= 5e+15) {
		tmp = fma(((double) M_PI), l_m, (tan((((double) M_PI) * l_m)) / (F * -F)));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 1e-7)
		tmp = fma(Float64(fma(l_m, Float64(l_m * Float64(0.3333333333333333 * Float64(pi * Float64(pi * pi)))), pi) / Float64(-F)), Float64(l_m / F), Float64(pi * l_m));
	elseif (Float64(pi * l_m) <= 5e+15)
		tmp = fma(pi, l_m, Float64(tan(Float64(pi * l_m)) / Float64(F * Float64(-F))));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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], 1e-7], N[(N[(N[(l$95$m * N[(l$95$m * N[(0.3333333333333333 * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision] / (-F)), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e+15], N[(Pi * l$95$m + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 9.9999999999999995e-8

   1. Initial program 83.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \mathsf{PI}\left(\right)\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)\right) \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\mathsf{PI}\left(\right)}^{3} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right), \mathsf{PI}\left(\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right) \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right), \mathsf{PI}\left(\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{PI}\left(\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      15. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{1}{3}}\right), \mathsf{PI}\left(\right)\right)\right) \]

      18. lower-PI.f64 65.3

          \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right), \color{blue}{\pi}\right)\right) \]

   5. Applied rewrites 65.3%

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right), \pi\right)\right)} \]

   7. Applied rewrites 71.9%

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

   if 9.9999999999999995e-8 < (*.f64 (PI.f64) l) < 5e15

   1. Initial program 100.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} + \left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)} \]

      12. lower-neg.f64 100.0

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

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{neg}\left(\color{blue}{\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{neg}\left(\color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{neg}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      16. un-div-inv N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{neg}\left(\color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot F}}\right)\right) \]

      17. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell \cdot \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi\right)}{-F}, \frac{\ell}{F}, \pi \cdot \ell\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot \left(-F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan \left(\pi \cdot l\_m\right)}{F}, \frac{-1}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 5e+15)
    (fma (/ (tan (* PI l_m)) F) (/ -1.0 F) (* PI l_m))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

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) <= 5e+15) {
		tmp = fma((tan((((double) M_PI) * l_m)) / F), (-1.0 / F), (((double) M_PI) * l_m));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 5e+15)
		tmp = fma(Float64(tan(Float64(pi * l_m)) / F), Float64(-1.0 / F), Float64(pi * l_m));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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], 5e+15], N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e15

   1. Initial program 83.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right) + \mathsf{PI}\left(\right) \cdot \ell} \]

   4. Applied rewrites 87.4%

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

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

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

Alternative 6: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 5e+15)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

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) <= 5e+15) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 5e+15) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = (l_m * Math.sqrt(Math.sqrt(Math.PI))) * Math.sqrt((Math.PI * Math.sqrt(Math.PI)));
	}
	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) <= 5e+15:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = (l_m * math.sqrt(math.sqrt(math.pi))) * math.sqrt((math.pi * math.sqrt(math.pi)))
	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) <= 5e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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) <= 5e+15)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = (l_m * sqrt(sqrt(pi))) * sqrt((pi * sqrt(pi)));
	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], 5e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e15

   1. Initial program 83.8%

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

      1. associate-/r* N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.3

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

   4. Applied rewrites 87.3%

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

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

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

Alternative 7: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{-1}{F \cdot \left(\mathsf{fma}\left(l\_m, \left(\pi \cdot l\_m\right) \cdot -0.3333333333333333, \frac{1}{\pi}\right) \cdot \frac{F}{l\_m}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 5e+15)
    (fma
     PI
     l_m
     (/
      -1.0
      (*
       F
       (* (fma l_m (* (* PI l_m) -0.3333333333333333) (/ 1.0 PI)) (/ F l_m)))))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

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) <= 5e+15) {
		tmp = fma(((double) M_PI), l_m, (-1.0 / (F * (fma(l_m, ((((double) M_PI) * l_m) * -0.3333333333333333), (1.0 / ((double) M_PI))) * (F / l_m)))));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 5e+15)
		tmp = fma(pi, l_m, Float64(-1.0 / Float64(F * Float64(fma(l_m, Float64(Float64(pi * l_m) * -0.3333333333333333), Float64(1.0 / pi)) * Float64(F / l_m)))));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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], 5e+15], N[(Pi * l$95$m + N[(-1.0 / N[(F * N[(N[(l$95$m * N[(N[(Pi * l$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[(F / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e15

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-tan.f64 N/A

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

      11. tan-quot N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}} \cdot \frac{\frac{1}{F}}{F} \]

      12. clear-num N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}}} \cdot \frac{\frac{1}{F}}{F} \]

      13. frac-times N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1 \cdot \frac{1}{F}}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F}} \]

      14. div-inv N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{1}{F}}}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F} \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\frac{1}{F}}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F}} \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{1}{F}}}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F} \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F}} \]

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\mathsf{PI}\left(\right)}}{\ell}} \cdot F} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\mathsf{PI}\left(\right)}}{\ell}} \cdot F} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\mathsf{PI}\left(\right)}}{\ell} \cdot F} \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{1}{\mathsf{PI}\left(\right)}}{\ell} \cdot F} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\color{blue}{\mathsf{fma}\left(\ell, \ell \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right), \frac{1}{\mathsf{PI}\left(\right)}\right)}}{\ell} \cdot F} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \color{blue}{\ell \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot F} \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right)}, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot F} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{3}}\right), \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot F} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{3}\right)}, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot F} \]

      9. lower-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{-1}{3}\right), \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot F} \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{3}\right), \color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{\ell} \cdot F} \]

      11. lower-PI.f64 90.4

          \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \left(\pi \cdot -0.3333333333333333\right), \frac{1}{\color{blue}{\pi}}\right)}{\ell} \cdot F} \]

   7. Applied rewrites 90.4%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{\mathsf{fma}\left(\ell, \ell \cdot \left(\pi \cdot -0.3333333333333333\right), \frac{1}{\pi}\right)}{\ell}} \cdot F} \]

   9. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \ell, \frac{-1}{F \cdot \left(\mathsf{fma}\left(\ell, -0.3333333333333333 \cdot \left(\ell \cdot \pi\right), \frac{1}{\pi}\right) \cdot \frac{F}{\ell}\right)}\right)} \]

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{-1}{F \cdot \left(\mathsf{fma}\left(\ell, \left(\pi \cdot \ell\right) \cdot -0.3333333333333333, \frac{1}{\pi}\right) \cdot \frac{F}{\ell}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(l\_m, l\_m \cdot \left(0.3333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \pi\right)}{-F}, \frac{l\_m}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 5e+15)
    (fma
     (/ (fma l_m (* l_m (* 0.3333333333333333 (* PI (* PI PI)))) PI) (- F))
     (/ l_m F)
     (* PI l_m))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

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) <= 5e+15) {
		tmp = fma((fma(l_m, (l_m * (0.3333333333333333 * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))), ((double) M_PI)) / -F), (l_m / F), (((double) M_PI) * l_m));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 5e+15)
		tmp = fma(Float64(fma(l_m, Float64(l_m * Float64(0.3333333333333333 * Float64(pi * Float64(pi * pi)))), pi) / Float64(-F)), Float64(l_m / F), Float64(pi * l_m));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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], 5e+15], N[(N[(N[(l$95$m * N[(l$95$m * N[(0.3333333333333333 * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision] / (-F)), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e15

   1. Initial program 83.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \mathsf{PI}\left(\right)\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)\right) \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{{\mathsf{PI}\left(\right)}^{3} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right), \mathsf{PI}\left(\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right) \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right), \mathsf{PI}\left(\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{PI}\left(\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\mathsf{PI}\left(\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}, \mathsf{PI}\left(\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      15. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      16. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{1}{3}}\right), \mathsf{PI}\left(\right)\right)\right) \]

      18. lower-PI.f64 65.0

          \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right), \color{blue}{\pi}\right)\right) \]

   5. Applied rewrites 65.0%

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right), \pi\right)\right)} \]

   7. Applied rewrites 71.5%

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

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

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

Alternative 9: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 5e+15)
    (- (* PI l_m) (/ (* PI (/ l_m F)) F))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

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) <= 5e+15) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 5e+15) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else {
		tmp = (l_m * Math.sqrt(Math.sqrt(Math.PI))) * Math.sqrt((Math.PI * Math.sqrt(Math.PI)));
	}
	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) <= 5e+15:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	else:
		tmp = (l_m * math.sqrt(math.sqrt(math.pi))) * math.sqrt((math.pi * math.sqrt(math.pi)))
	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) <= 5e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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) <= 5e+15)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	else
		tmp = (l_m * sqrt(sqrt(pi))) * sqrt((pi * sqrt(pi)));
	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], 5e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e15

   1. Initial program 83.8%

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

      1. associate-/r* N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.3

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\color{blue}{\ell \cdot \mathsf{PI}\left(\right)}}{F}}{F} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 80.3

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

   7. Applied rewrites 80.3%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.4

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

   9. Applied rewrites 80.4%

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

   if 5e15 < (*.f64 (PI.f64) l)

   1. Initial program 60.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \ell \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \]

      4. rem-square-sqrt N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \mathsf{PI}\left(\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \mathsf{PI}\left(\right)} \]

      7. associate-*l* N/A

         \[\leadsto \ell \cdot \sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \]

      10. sqrt-unprod N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \ell \cdot \left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \ell\right) \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   7. Applied rewrites 99.6%

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

Alternative 10: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 20000000000000.0)
    (- (* PI l_m) (/ (* PI (/ l_m F)) F))
    (* PI l_m))))

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) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	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) <= 20000000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	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) <= 20000000000000.0:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	else:
		tmp = math.pi * l_m
	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) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(pi * l_m);
	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) <= 20000000000000.0)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	else
		tmp = pi * l_m;
	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], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e13

   1. Initial program 83.7%

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

      1. associate-/r* N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.3

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\color{blue}{\ell \cdot \mathsf{PI}\left(\right)}}{F}}{F} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 80.2

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

   7. Applied rewrites 80.2%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 80.3

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

   9. Applied rewrites 80.3%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\frac{\pi}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 20000000000000.0)
    (* l_m (- PI (/ (/ PI F) F)))
    (* PI l_m))))

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) <= 20000000000000.0) {
		tmp = l_m * (((double) M_PI) - ((((double) M_PI) / F) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	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) <= 20000000000000.0) {
		tmp = l_m * (Math.PI - ((Math.PI / F) / F));
	} else {
		tmp = Math.PI * l_m;
	}
	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) <= 20000000000000.0:
		tmp = l_m * (math.pi - ((math.pi / F) / F))
	else:
		tmp = math.pi * l_m
	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) <= 20000000000000.0)
		tmp = Float64(l_m * Float64(pi - Float64(Float64(pi / F) / F)));
	else
		tmp = Float64(pi * l_m);
	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) <= 20000000000000.0)
		tmp = l_m * (pi - ((pi / F) / F));
	else
		tmp = pi * l_m;
	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], 20000000000000.0], N[(l$95$m * N[(Pi - N[(N[(Pi / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e13

   1. Initial program 83.7%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      3. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

      1. lift-PI.f64 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\frac{\pi}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 20000000000000.0)
    (* l_m (- PI (/ PI (* F F))))
    (* PI l_m))))

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) <= 20000000000000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	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) <= 20000000000000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	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) <= 20000000000000.0:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	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) <= 20000000000000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	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) <= 20000000000000.0)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	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], 20000000000000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e13

   1. Initial program 83.7%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      3. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 61.5%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.8% accurate, 22.5× speedup

Math

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

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)))

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);
}

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);
}

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)

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(pi * 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);
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[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in l around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

   2. lower-PI.f64 74.0

      \[\leadsto \ell \cdot \color{blue}{\pi} \]

5. Applied rewrites 74.0%

   \[\leadsto \color{blue}{\ell \cdot \pi} \]

6. Final simplification 74.0%

   \[\leadsto \pi \cdot \ell \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))