VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 99.3%

Time: 4.9s

Alternatives: 6

Speedup: N/A×

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: 76.1% 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: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{{F}^{-1} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot \sin \left(\mathsf{fma}\left(\pi, l\_m, \frac{-1 \cdot \pi}{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 4.2e+15)
    (-
     (* PI l_m)
     (/
      (* (pow F -1.0) (sin (* PI l_m)))
      (* F (sin (fma PI l_m (/ (* -1.0 PI) -2.0))))))
    (* 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 (l_m <= 4.2e+15) {
		tmp = (((double) M_PI) * l_m) - ((pow(F, -1.0) * sin((((double) M_PI) * l_m))) / (F * sin(fma(((double) M_PI), l_m, ((-1.0 * ((double) M_PI)) / -2.0)))));
	} else {
		tmp = ((double) M_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 (l_m <= 4.2e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64((F ^ -1.0) * sin(Float64(pi * l_m))) / Float64(F * sin(fma(pi, l_m, Float64(Float64(-1.0 * pi) / -2.0))))));
	else
		tmp = Float64(pi * l_m);
	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[l$95$m, 4.2e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Power[F, -1.0], $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * N[Sin[N[(Pi * l$95$m + N[(N[(-1.0 * Pi), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.2e15

   1. Initial program 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. sin-+PI/2-rev N/A

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

      21. lower-sin.f64 N/A

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

   4. Applied rewrites 90.5%

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

   if 4.2e15 < l

   1. Initial program 63.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

Alternative 2: 98.5% accurate, N/A× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\left({F}^{-0.5} \cdot {F}^{-0.5}\right) \cdot \sin \left(\pi \cdot l\_m\right)}{\left(-1 \cdot F\right) \cdot \sin \left(\mathsf{fma}\left(\pi, l\_m, \frac{-1 \cdot \pi}{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 4.2e+15)
    (+
     (* PI l_m)
     (/
      (* (* (pow F -0.5) (pow F -0.5)) (sin (* PI l_m)))
      (* (* -1.0 F) (sin (fma PI l_m (/ (* -1.0 PI) -2.0))))))
    (* 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 (l_m <= 4.2e+15) {
		tmp = (((double) M_PI) * l_m) + (((pow(F, -0.5) * pow(F, -0.5)) * sin((((double) M_PI) * l_m))) / ((-1.0 * F) * sin(fma(((double) M_PI), l_m, ((-1.0 * ((double) M_PI)) / -2.0)))));
	} else {
		tmp = ((double) M_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 (l_m <= 4.2e+15)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64((F ^ -0.5) * (F ^ -0.5)) * sin(Float64(pi * l_m))) / Float64(Float64(-1.0 * F) * sin(fma(pi, l_m, Float64(Float64(-1.0 * pi) / -2.0))))));
	else
		tmp = Float64(pi * l_m);
	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[l$95$m, 4.2e+15], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(N[Power[F, -0.5], $MachinePrecision] * N[Power[F, -0.5], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 * F), $MachinePrecision] * N[Sin[N[(Pi * l$95$m + N[(N[(-1.0 * Pi), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.2e15

   1. Initial program 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. sin-+PI/2-rev N/A

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

      21. lower-sin.f64 N/A

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

   4. Applied rewrites 90.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{{F}^{-1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \sin \left(\mathsf{fma}\left(\pi, \ell, \frac{-1 \cdot \pi}{-2}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 46.3

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

   6. Applied rewrites 46.3%

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

   if 4.2e15 < l

   1. Initial program 63.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \ell + \frac{\left({F}^{-0.5} \cdot {F}^{-0.5}\right) \cdot \sin \left(\pi \cdot \ell\right)}{\left(-1 \cdot F\right) \cdot \sin \left(\mathsf{fma}\left(\pi, \ell, \frac{-1 \cdot \pi}{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, N/A× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F \cdot \sin \left(\mathsf{fma}\left(\pi, l\_m, \frac{-1 \cdot \pi}{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 4.4e+15)
    (-
     (* PI l_m)
     (/ (* l_m (/ PI F)) (* F (sin (fma PI l_m (/ (* -1.0 PI) -2.0))))))
    (* 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 (l_m <= 4.4e+15) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / (F * sin(fma(((double) M_PI), l_m, ((-1.0 * ((double) M_PI)) / -2.0)))));
	} else {
		tmp = ((double) M_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 (l_m <= 4.4e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / Float64(F * sin(fma(pi, l_m, Float64(Float64(-1.0 * pi) / -2.0))))));
	else
		tmp = Float64(pi * l_m);
	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[l$95$m, 4.4e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[Sin[N[(Pi * l$95$m + N[(N[(-1.0 * Pi), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.4e15

   1. Initial program 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. sin-+PI/2-rev N/A

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

      21. lower-sin.f64 N/A

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if 4.4e15 < l

   1. Initial program 63.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

Alternative 4: 92.0% accurate, N/A× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\sin \left(\pi \cdot l\_m\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\pi, l\_m, \frac{-1 \cdot \pi}{-2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 4.2e+15)
    (-
     (* PI l_m)
     (/ (/ (sin (* PI l_m)) (* F F)) (sin (fma PI l_m (/ (* -1.0 PI) -2.0)))))
    (* 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 (l_m <= 4.2e+15) {
		tmp = (((double) M_PI) * l_m) - ((sin((((double) M_PI) * l_m)) / (F * F)) / sin(fma(((double) M_PI), l_m, ((-1.0 * ((double) M_PI)) / -2.0))));
	} else {
		tmp = ((double) M_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 (l_m <= 4.2e+15)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(sin(Float64(pi * l_m)) / Float64(F * F)) / sin(fma(pi, l_m, Float64(Float64(-1.0 * pi) / -2.0)))));
	else
		tmp = Float64(pi * l_m);
	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[l$95$m, 4.2e+15], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] / N[Sin[N[(Pi * l$95$m + N[(N[(-1.0 * Pi), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.2e15

   1. Initial program 84.6%

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

   3. Taylor expanded in F around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. *-commutative N/A

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{{F}^{2}}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{{F}^{2}}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      7. lift-PI.f64 N/A

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2}}}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} \]

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. sin-+PI/2-rev N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\ell \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\ell \cdot \mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{PI}\left(\right) \cdot \ell + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      14. lift-PI.f64 N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\pi, \ell, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      15. frac-2neg N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\pi, \ell, \frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(2\right)}\right)\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\pi, \ell, \frac{-1 \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(2\right)}\right)\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\pi, \ell, \frac{-1 \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(2\right)}\right)\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}}{\sin \left(\mathsf{fma}\left(\pi, \ell, \frac{-1 \cdot \mathsf{PI}\left(\right)}{\mathsf{neg}\left(2\right)}\right)\right)} \]

      19. lift-PI.f64 N/A

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

      20. metadata-eval 85.5

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

   5. Applied rewrites 85.5%

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

   if 4.2e15 < l

   1. Initial program 63.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

Alternative 5: 74.2% accurate, N/A× speedup

Math

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

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 (<= l_m 0.5) (* (- PI (/ PI (* F F))) l_m) (* 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 (l_m <= 0.5) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * l_m;
	} 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 (l_m <= 0.5) {
		tmp = (Math.PI - (Math.PI / (F * F))) * l_m;
	} 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 l_m <= 0.5:
		tmp = (math.pi - (math.pi / (F * F))) * l_m
	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 (l_m <= 0.5)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * l_m);
	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 (l_m <= 0.5)
		tmp = (pi - (pi / (F * F))) * l_m;
	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[l$95$m, 0.5], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 84.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

         \[\leadsto \left(\pi - \frac{\pi}{{F}^{2}}\right) \cdot \ell \]

      7. pow2 N/A

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

      8. lift-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if 0.5 < l

   1. Initial program 63.3%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 97.8

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 97.8%

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

Alternative 6: 73.9% accurate, N/A× 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 80.3%

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

3. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 76.7

      \[\leadsto \pi \cdot \ell \]

5. Applied rewrites 76.7%

   \[\leadsto \color{blue}{\pi \cdot \ell} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))