VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.7% → 99.4%

Time: 24.1s

Alternatives: 9

Speedup: 8.1×

• could not determine a ground truth

  1. F = 5.445753058088837e-162
  2. l = -1913044565503538.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\sin \left(\pi \cdot l\_m\right)}{F}}{F \cdot \cos \left(\pi \cdot l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= (* PI l_m) 2e+14)
    (- (* PI l_m) (/ (/ (sin (* PI l_m)) F) (* 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 ((((double) M_PI) * l_m) <= 2e+14) {
		tmp = (((double) M_PI) * l_m) - ((sin((((double) M_PI) * l_m)) / F) / (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 ((Math.PI * l_m) <= 2e+14) {
		tmp = (Math.PI * l_m) - ((Math.sin((Math.PI * l_m)) / F) / (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 (math.pi * l_m) <= 2e+14:
		tmp = (math.pi * l_m) - ((math.sin((math.pi * l_m)) / F) / (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 (Float64(pi * l_m) <= 2e+14)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(sin(Float64(pi * l_m)) / F) / 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 ((pi * l_m) <= 2e+14)
		tmp = (pi * l_m) - ((sin((pi * l_m)) / F) / (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[N[(Pi * l$95$m), $MachinePrecision], 2e+14], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $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 (*.f64 (PI.f64) l) < 2e14

   1. Initial program 79.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot \frac{\color{blue}{1}}{F \cdot F}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot \frac{\frac{1}{F}}{\color{blue}{F}}\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \frac{1}{F}}{\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot F}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \frac{1}{F}\right), \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot F\right)}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right), F\right), \left(\color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right), F\right), \left(\cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)} \cdot F\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right), F\right), \left(\cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \ell\right) \cdot F\right)\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), \left(\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot F\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), \mathsf{*.f64}\left(\cos \left(\mathsf{PI}\left(\right) \cdot \ell\right), \color{blue}{F}\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right), F\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right), F\right)\right)\right) \]

      14. PI-lowering-PI.f64 88.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right)\right)\right) \]

   4. Applied egg-rr 88.6%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 56.4%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 99.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e+14)
    (- (* PI l_m) (/ (/ (tan (* 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) <= 2e+14) {
		tmp = (((double) M_PI) * l_m) - ((tan((((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) <= 2e+14) {
		tmp = (Math.PI * l_m) - ((Math.tan((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) <= 2e+14:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e+14)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 2e+14)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e+14], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 79.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\right), \color{blue}{F}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), F\right), F\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right), F\right), F\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

      8. PI-lowering-PI.f64 88.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

   4. Applied egg-rr 88.6%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 56.4%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 99.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{1}{F}}{\frac{\frac{\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right)\right) \cdot \left(F \cdot \left(l\_m \cdot l\_m\right)\right)}{\pi \cdot \pi} - \frac{F}{\pi}}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= (* PI l_m) 2e+14)
    (+
     (* PI l_m)
     (/
      (/ 1.0 F)
      (/
       (-
        (/
         (* (* PI (* (* PI PI) 0.3333333333333333)) (* F (* l_m l_m)))
         (* PI PI))
        (/ F 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 ((((double) M_PI) * l_m) <= 2e+14) {
		tmp = (((double) M_PI) * l_m) + ((1.0 / F) / (((((((double) M_PI) * ((((double) M_PI) * ((double) M_PI)) * 0.3333333333333333)) * (F * (l_m * l_m))) / (((double) M_PI) * ((double) M_PI))) - (F / ((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 ((Math.PI * l_m) <= 2e+14) {
		tmp = (Math.PI * l_m) + ((1.0 / F) / (((((Math.PI * ((Math.PI * Math.PI) * 0.3333333333333333)) * (F * (l_m * l_m))) / (Math.PI * Math.PI)) - (F / 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 (math.pi * l_m) <= 2e+14:
		tmp = (math.pi * l_m) + ((1.0 / F) / (((((math.pi * ((math.pi * math.pi) * 0.3333333333333333)) * (F * (l_m * l_m))) / (math.pi * math.pi)) - (F / 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 (Float64(pi * l_m) <= 2e+14)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(1.0 / F) / Float64(Float64(Float64(Float64(Float64(pi * Float64(Float64(pi * pi) * 0.3333333333333333)) * Float64(F * Float64(l_m * l_m))) / Float64(pi * pi)) - Float64(F / 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 ((pi * l_m) <= 2e+14)
		tmp = (pi * l_m) + ((1.0 / F) / (((((pi * ((pi * pi) * 0.3333333333333333)) * (F * (l_m * l_m))) / (pi * pi)) - (F / 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[N[(Pi * l$95$m), $MachinePrecision], 2e+14], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(1.0 / F), $MachinePrecision] / N[(N[(N[(N[(N[(Pi * N[(N[(Pi * Pi), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(F * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[(F / Pi), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 79.1%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1 \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1}{F} \cdot \color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1}{F} \cdot \frac{1}{\color{blue}{\frac{F}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}}}\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{1}{F}}{\color{blue}{\frac{F}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{1}{F}\right), \color{blue}{\left(\frac{F}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \left(\frac{\color{blue}{F}}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right)\right) \]

      8. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \mathsf{tan.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 88.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \color{blue}{\left(\frac{-1 \cdot \frac{F \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}{{\mathsf{PI}\left(\right)}^{2}} + \frac{F}{\mathsf{PI}\left(\right)}}{\ell}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{F \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}{{\mathsf{PI}\left(\right)}^{2}} + \frac{F}{\mathsf{PI}\left(\right)}\right), \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 94.9%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 56.4%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{1}{F}}{\frac{\frac{\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\pi \cdot \pi} - \frac{F}{\pi}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{l\_m}{\frac{\left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right) \cdot \left(F \cdot \left(l\_m \cdot l\_m\right)\right) - F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 6e+14)
    (+
     (* PI l_m)
     (/
      (/
       l_m
       (/ (- (* (* (* PI PI) 0.3333333333333333) (* F (* l_m l_m))) F) PI))
      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 <= 6e+14) {
		tmp = (((double) M_PI) * l_m) + ((l_m / (((((((double) M_PI) * ((double) M_PI)) * 0.3333333333333333) * (F * (l_m * l_m))) - F) / ((double) M_PI))) / 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 <= 6e+14) {
		tmp = (Math.PI * l_m) + ((l_m / (((((Math.PI * Math.PI) * 0.3333333333333333) * (F * (l_m * l_m))) - F) / Math.PI)) / 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 <= 6e+14:
		tmp = (math.pi * l_m) + ((l_m / (((((math.pi * math.pi) * 0.3333333333333333) * (F * (l_m * l_m))) - F) / math.pi)) / 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 <= 6e+14)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(l_m / Float64(Float64(Float64(Float64(Float64(pi * pi) * 0.3333333333333333) * Float64(F * Float64(l_m * l_m))) - F) / pi)) / 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 <= 6e+14)
		tmp = (pi * l_m) + ((l_m / (((((pi * pi) * 0.3333333333333333) * (F * (l_m * l_m))) - F) / pi)) / 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, 6e+14], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(l$95$m / N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(F * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - F), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6e14

   1. Initial program 79.1%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1 \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1}{F} \cdot \color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1}{F} \cdot \frac{1}{\color{blue}{\frac{F}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}}}\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{1}{F}}{\color{blue}{\frac{F}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{1}{F}\right), \color{blue}{\left(\frac{F}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \left(\frac{\color{blue}{F}}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right)\right) \]

      8. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \mathsf{tan.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 88.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(F, \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \color{blue}{\left(\frac{-1 \cdot \frac{F \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}{{\mathsf{PI}\left(\right)}^{2}} + \frac{F}{\mathsf{PI}\left(\right)}}{\ell}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{F \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}{{\mathsf{PI}\left(\right)}^{2}} + \frac{F}{\mathsf{PI}\left(\right)}\right), \color{blue}{\ell}\right)\right)\right) \]

   7. Simplified 94.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{\frac{F}{\pi} - \frac{\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\pi \cdot \pi}}{\ell}}} \]
   8. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right), \color{blue}{\left(\frac{\frac{1}{F}}{\frac{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{\ell}}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right), \left(\frac{\color{blue}{\frac{1}{F}}}{\frac{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{\ell}}\right)\right) \]

      3. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\color{blue}{1}}{F}}{\frac{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{\ell}}\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{1}{\color{blue}{\frac{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{\ell} \cdot F}}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{1}{\frac{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{\ell}}}{\color{blue}{F}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\ell}{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}}{F}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{F}{\mathsf{PI}\left(\right)} - \frac{\left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right)\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right), \color{blue}{F}\right)\right) \]

   9. Applied egg-rr 94.9%

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

   if 6e14 < l

   1. Initial program 56.4%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{\ell}{\frac{\left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right) \cdot \left(F \cdot \left(\ell \cdot \ell\right)\right) - F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;F \cdot F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\pi \cdot l\_m}{0 - F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= (* F F) 5e-317)
    (* PI l_m)
    (if (<= (* F F) 2.5e-9) (/ (* PI l_m) (- 0.0 (* 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 ((F * F) <= 5e-317) {
		tmp = ((double) M_PI) * l_m;
	} else if ((F * F) <= 2.5e-9) {
		tmp = (((double) M_PI) * l_m) / (0.0 - (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 ((F * F) <= 5e-317) {
		tmp = Math.PI * l_m;
	} else if ((F * F) <= 2.5e-9) {
		tmp = (Math.PI * l_m) / (0.0 - (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 (F * F) <= 5e-317:
		tmp = math.pi * l_m
	elif (F * F) <= 2.5e-9:
		tmp = (math.pi * l_m) / (0.0 - (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(F * F) <= 5e-317)
		tmp = Float64(pi * l_m);
	elseif (Float64(F * F) <= 2.5e-9)
		tmp = Float64(Float64(pi * l_m) / Float64(0.0 - 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 ((F * F) <= 5e-317)
		tmp = pi * l_m;
	elseif ((F * F) <= 2.5e-9)
		tmp = (pi * l_m) / (0.0 - (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[(F * F), $MachinePrecision], 5e-317], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[N[(F * F), $MachinePrecision], 2.5e-9], N[(N[(Pi * l$95$m), $MachinePrecision] / N[(0.0 - N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 5.00000017e-317 or 2.5000000000000001e-9 < (*.f64 F F)

   1. Initial program 73.6%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 80.4%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 80.4%

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

   if 5.00000017e-317 < (*.f64 F F) < 2.5000000000000001e-9

   1. Initial program 76.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\right), \color{blue}{F}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), F\right), F\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right), F\right), F\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

      8. PI-lowering-PI.f64 78.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

   4. Applied egg-rr 78.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}, F\right), F\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI}\left(\right)\right), F\right), F\right)\right) \]

      2. PI-lowering-PI.f64 58.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), F\right), F\right)\right) \]

   7. Simplified 58.0%

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

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   9. 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. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(\mathsf{neg}\left({F}^{2}\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI}\left(\right)\right), \left(\mathsf{neg}\left(\color{blue}{{F}^{2}}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), \left(\mathsf{neg}\left({F}^{\color{blue}{2}}\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\left({F}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\left(F \cdot F\right)\right)\right) \]

      8. *-lowering-*.f64 57.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(F, F\right)\right)\right) \]

   10. Simplified 57.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \cdot F \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\pi \cdot \ell}{0 - F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 27500000:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 27500000.0) (- (* PI l_m) (/ (* 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 (l_m <= 27500000.0) {
		tmp = (((double) M_PI) * l_m) - ((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 (l_m <= 27500000.0) {
		tmp = (Math.PI * l_m) - ((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 l_m <= 27500000.0:
		tmp = (math.pi * l_m) - ((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 (l_m <= 27500000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 27500000.0)
		tmp = (pi * l_m) - ((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[l$95$m, 27500000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.75e7

   1. Initial program 79.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\right), \color{blue}{F}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), F\right), F\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right), F\right), F\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

      8. PI-lowering-PI.f64 88.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}, F\right), F\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI}\left(\right)\right), F\right), F\right)\right) \]

      2. PI-lowering-PI.f64 84.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), F\right), F\right)\right) \]

   7. Simplified 84.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}\right), F\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{F} \cdot \ell\right), F\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{F}\right), \ell\right), F\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), F\right), \ell\right), F\right)\right) \]

      5. PI-lowering-PI.f64 84.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), F\right), \ell\right), F\right)\right) \]

   9. Applied egg-rr 84.4%

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

   if 2.75e7 < l

   1. Initial program 57.9%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 94.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 94.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 27500000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 27500000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 27500000.0) (- (* PI l_m) (* (/ PI F) (/ 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 <= 27500000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (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 <= 27500000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (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 <= 27500000.0:
		tmp = (math.pi * l_m) - ((math.pi / F) * (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 <= 27500000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(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 <= 27500000.0)
		tmp = (pi * l_m) - ((pi / F) * (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, 27500000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.75e7

   1. Initial program 79.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\right), \color{blue}{F}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), F\right), F\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \ell\right)\right), F\right), F\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

      8. PI-lowering-PI.f64 88.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right), F\right), F\right)\right) \]

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}, F\right), F\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI}\left(\right)\right), F\right), F\right)\right) \]

      2. PI-lowering-PI.f64 84.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), F\right), F\right)\right) \]

   7. Simplified 84.4%

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

      1. associate-/l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\mathsf{PI}\left(\right) \cdot \ell}{\color{blue}{F} \cdot F}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\mathsf{PI}\left(\right)}{F} \cdot \color{blue}{\frac{\ell}{F}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{F}\right), \color{blue}{\left(\frac{\ell}{F}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), F\right), \left(\frac{\color{blue}{\ell}}{F}\right)\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), F\right), \left(\frac{\ell}{F}\right)\right)\right) \]

      7. /-lowering-/.f64 84.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), F\right), \mathsf{/.f64}\left(\ell, \color{blue}{F}\right)\right)\right) \]

   9. Applied egg-rr 84.4%

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

   if 2.75e7 < l

   1. Initial program 57.9%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 94.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 94.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 27500000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 27500000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 27500000.0) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 27500000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 27500000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 27500000.0:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 27500000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 27500000.0)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 27500000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.75e7

   1. Initial program 79.1%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)}\right)\right) \]

      3. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({F}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{F}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(F \cdot \color{blue}{F}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 74.2%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(F, \color{blue}{F}\right)\right)\right)\right) \]

   5. Simplified 74.2%

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

   if 2.75e7 < l

   1. Initial program 57.9%

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

   3. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. PI-lowering-PI.f64 94.5%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

   5. Simplified 94.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 27500000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.6% accurate, 37.7× 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 74.4%

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

3. Taylor expanded in l around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. PI-lowering-PI.f64 70.3%

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right) \]

5. Simplified 70.3%

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

6. Final simplification 70.3%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))