VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.2%

Time: 4.4s

Alternatives: 12

Speedup: 3.2×

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: 76.9% 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, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 10500000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot \sin \left(\left(-\pi \cdot l\_m\right) + \frac{\pi}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 10500000000000.0)
    (-
     (* PI l_m)
     (/
      (* (/ 1.0 F) (sin (* PI l_m)))
      (* F (sin (+ (- (* PI l_m)) (/ 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 <= 10500000000000.0) {
		tmp = (((double) M_PI) * l_m) - (((1.0 / F) * sin((((double) M_PI) * l_m))) / (F * sin((-(((double) M_PI) * l_m) + (((double) M_PI) / 2.0)))));
	} 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 <= 10500000000000.0) {
		tmp = (Math.PI * l_m) - (((1.0 / F) * Math.sin((Math.PI * l_m))) / (F * Math.sin((-(Math.PI * l_m) + (Math.PI / 2.0)))));
	} 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 <= 10500000000000.0:
		tmp = (math.pi * l_m) - (((1.0 / F) * math.sin((math.pi * l_m))) / (F * math.sin((-(math.pi * l_m) + (math.pi / 2.0)))))
	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 <= 10500000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(1.0 / F) * sin(Float64(pi * l_m))) / Float64(F * sin(Float64(Float64(-Float64(pi * l_m)) + Float64(pi / 2.0))))));
	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 <= 10500000000000.0)
		tmp = (pi * l_m) - (((1.0 / F) * sin((pi * l_m))) / (F * sin((-(pi * l_m) + (pi / 2.0)))));
	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, 10500000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(1.0 / F), $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * N[Sin[N[((-N[(Pi * l$95$m), $MachinePrecision]) + N[(Pi / 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 < 1.05e13

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-sin.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if 1.05e13 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 10500000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot \sin \left(\mathsf{fma}\left(\pi, l\_m, \frac{\pi}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 10500000000000.0)
    (-
     (* PI l_m)
     (/ (* (/ 1.0 F) (sin (* PI l_m))) (* F (sin (fma PI l_m (/ 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 <= 10500000000000.0) {
		tmp = (((double) M_PI) * l_m) - (((1.0 / F) * sin((((double) M_PI) * l_m))) / (F * sin(fma(((double) M_PI), l_m, (((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 <= 10500000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(1.0 / F) * sin(Float64(pi * l_m))) / Float64(F * sin(fma(pi, l_m, Float64(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, 10500000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(1.0 / F), $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * N[Sin[N[(Pi * l$95$m + N[(Pi / 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 < 1.05e13

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-PI.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1.05e13 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 10500000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot \cos \left(\pi \cdot l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 10500000000000.0)
    (- (* PI l_m) (/ (* (/ 1.0 F) (sin (* PI l_m))) (* F (cos (* PI 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 <= 10500000000000.0) {
		tmp = (((double) M_PI) * l_m) - (((1.0 / F) * sin((((double) M_PI) * l_m))) / (F * cos((((double) M_PI) * 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 <= 10500000000000.0) {
		tmp = (Math.PI * l_m) - (((1.0 / F) * Math.sin((Math.PI * l_m))) / (F * Math.cos((Math.PI * 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 <= 10500000000000.0:
		tmp = (math.pi * l_m) - (((1.0 / F) * math.sin((math.pi * l_m))) / (F * math.cos((math.pi * 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 <= 10500000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(1.0 / F) * sin(Float64(pi * l_m))) / Float64(F * cos(Float64(pi * 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 <= 10500000000000.0)
		tmp = (pi * l_m) - (((1.0 / F) * sin((pi * l_m))) / (F * cos((pi * 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, 10500000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(1.0 / F), $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.05e13

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

   if 1.05e13 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 10500000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot l\_m\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 10500000000000.0)
    (- (* PI l_m) (* (/ 1.0 F) (/ (tan (* PI l_m)) 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 (l_m <= 10500000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((1.0 / F) * (tan((((double) M_PI) * l_m)) / 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 (l_m <= 10500000000000.0) {
		tmp = (Math.PI * l_m) - ((1.0 / F) * (Math.tan((Math.PI * l_m)) / 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 l_m <= 10500000000000.0:
		tmp = (math.pi * l_m) - ((1.0 / F) * (math.tan((math.pi * l_m)) / 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 (l_m <= 10500000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / F) * Float64(tan(Float64(pi * l_m)) / 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 (l_m <= 10500000000000.0)
		tmp = (pi * l_m) - ((1.0 / F) * (tan((pi * l_m)) / 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[l$95$m, 10500000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] * N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.05e13

   1. Initial program 76.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. 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. pow2 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. inv-pow N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 82.4

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

   3. Applied rewrites 82.4%

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

   if 1.05e13 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 5: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8500000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 8500000000000.0)
    (- (* PI l_m) (/ (* (/ 1.0 F) (sin (* PI l_m))) (* F 1.0)))
    (* 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 <= 8500000000000.0) {
		tmp = (((double) M_PI) * l_m) - (((1.0 / F) * sin((((double) M_PI) * l_m))) / (F * 1.0));
	} 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 <= 8500000000000.0) {
		tmp = (Math.PI * l_m) - (((1.0 / F) * Math.sin((Math.PI * l_m))) / (F * 1.0));
	} 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 <= 8500000000000.0:
		tmp = (math.pi * l_m) - (((1.0 / F) * math.sin((math.pi * l_m))) / (F * 1.0))
	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 <= 8500000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(1.0 / F) * sin(Float64(pi * l_m))) / Float64(F * 1.0)));
	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 <= 8500000000000.0)
		tmp = (pi * l_m) - (((1.0 / F) * sin((pi * l_m))) / (F * 1.0));
	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, 8500000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(1.0 / F), $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.5e12

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 81.6%

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

   if 8.5e12 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 6: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8500000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{F} \cdot \frac{\sin \left(\pi \cdot l\_m\right)}{1 \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 8500000000000.0)
    (- (* PI l_m) (* (/ 1.0 F) (/ (sin (* PI l_m)) (* 1.0 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 (l_m <= 8500000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((1.0 / F) * (sin((((double) M_PI) * l_m)) / (1.0 * 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 (l_m <= 8500000000000.0) {
		tmp = (Math.PI * l_m) - ((1.0 / F) * (Math.sin((Math.PI * l_m)) / (1.0 * 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 l_m <= 8500000000000.0:
		tmp = (math.pi * l_m) - ((1.0 / F) * (math.sin((math.pi * l_m)) / (1.0 * 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 (l_m <= 8500000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / F) * Float64(sin(Float64(pi * l_m)) / Float64(1.0 * 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 (l_m <= 8500000000000.0)
		tmp = (pi * l_m) - ((1.0 / F) * (sin((pi * l_m)) / (1.0 * 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[l$95$m, 8500000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] * N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(1.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.5e12

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-sin.f64 81.6

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

   8. Applied rewrites 81.6%

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

   if 8.5e12 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 7: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 9000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F \cdot \mathsf{fma}\left(-0.5 \cdot \left(l\_m \cdot l\_m\right), \pi \cdot \pi, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 9000000000000.0)
    (-
     (* PI l_m)
     (/ (* l_m (/ PI F)) (* F (fma (* -0.5 (* l_m l_m)) (* PI PI) 1.0))))
    (* 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 <= 9000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / (F * fma((-0.5 * (l_m * l_m)), (((double) M_PI) * ((double) M_PI)), 1.0)));
	} 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 <= 9000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / Float64(F * fma(Float64(-0.5 * Float64(l_m * l_m)), Float64(pi * pi), 1.0))));
	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, 9000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[(N[(-0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 9e12

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 81.6%

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

   7. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 75.3

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

   9. Applied rewrites 75.3%

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

   10. Taylor expanded in l around 0

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

       1. cos-neg-rev N/A

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

       2. sin-+PI/2 N/A

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lift-PI.f64 81.7

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

   12. Applied rewrites 81.7%

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

   if 9e12 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 8: 98.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4800000:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4800000.0)
    (- (* PI l_m) (/ (* l_m (/ PI F)) (* F 1.0)))
    (* 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 <= 4800000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / (F * 1.0));
	} 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 <= 4800000.0) {
		tmp = (Math.PI * l_m) - ((l_m * (Math.PI / F)) / (F * 1.0));
	} 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 <= 4800000.0:
		tmp = (math.pi * l_m) - ((l_m * (math.pi / F)) / (F * 1.0))
	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 <= 4800000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / Float64(F * 1.0)));
	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 <= 4800000.0)
		tmp = (pi * l_m) - ((l_m * (pi / F)) / (F * 1.0));
	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, 4800000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / N[(F * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.8e6

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. *-commutative N/A

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

      10. quot-tan N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-sin.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 82.4%

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

   4. Taylor expanded in l around 0

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

   6. Applied rewrites 81.6%

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

   7. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 75.3

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

   9. Applied rewrites 75.3%

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

   if 4.8e6 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 9: 92.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4800000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 4800000.0) (* (- 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 <= 4800000.0) {
		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 <= 4800000.0) {
		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 <= 4800000.0:
		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 <= 4800000.0)
		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 <= 4800000.0)
		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, 4800000.0], 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 < 4.8e6

   1. Initial program 76.9%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   3. 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 69.7

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

   4. Applied rewrites 69.7%

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

   if 4.8e6 < l

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 10: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -4 \cdot 10^{-164}:\\ \;\;\;\;\frac{\left(-\pi\right) \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -4e-164)
    (/ (* (- 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -4e-164) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -4e-164) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -4e-164:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -4e-164)
		tmp = Float64(Float64(Float64(-pi) * l_m) / 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) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -4e-164)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-164], 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)))) < -3.99999999999999985e-164

   1. Initial program 76.9%

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

   2. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 22.7

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

         \[\leadsto -\ell \cdot \frac{\pi}{{F}^{2}} \]

      7. pow2 N/A

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

      8. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 21.4

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

   9. Applied rewrites 21.4%

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

   if -3.99999999999999985e-164 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 11: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -4 \cdot 10^{-164}:\\ \;\;\;\;-l\_m \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -4e-164)
    (- (* 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -4e-164) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -4e-164) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -4e-164:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -4e-164)
		tmp = Float64(-Float64(l_m * 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) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -4e-164)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-164], (-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)))) < -3.99999999999999985e-164

   1. Initial program 76.9%

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

   2. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 22.7

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

         \[\leadsto -\ell \cdot \frac{\pi}{{F}^{2}} \]

      7. pow2 N/A

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

      8. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   if -3.99999999999999985e-164 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 12: 73.6% accurate, 13.6× 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 76.9%

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

2. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. 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 73.6

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

4. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))