VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 98.6%

Time: 21.1s

Alternatives: 7

Speedup: 3.7×

• could not determine a ground truth

  1. F = -2.1083914206568767e+138
  2. l = 2361180738048065.5

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 50000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 50000.0)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* (* l_m (* PI PI)) (/ 1.0 PI)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 50000.0) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (l_m * (((double) M_PI) * ((double) M_PI))) * (1.0 / ((double) M_PI));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 50000.0) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = (l_m * (Math.PI * Math.PI)) * (1.0 / Math.PI);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 50000.0:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = (l_m * (math.pi * math.pi)) * (1.0 / math.pi)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 50000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(l_m * Float64(pi * pi)) * Float64(1.0 / pi));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 50000.0)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = (l_m * (pi * pi)) * (1.0 / pi);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 50000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 80.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 91.1

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

   4. Applied egg-rr 91.1%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.6

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. *-rgt-identity N/A

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

      2. rgt-mult-inverse N/A

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

      3. clear-num N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)} \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \frac{\ell}{\mathsf{PI}\left(\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \ell}{\mathsf{PI}\left(\right)}} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\ell}\right)} \cdot \ell}{\mathsf{PI}\left(\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{\ell} \cdot \ell\right)}}{\mathsf{PI}\left(\right)} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \mathsf{fma}\left(\pi, \ell, 0\right), 0\right) \cdot \frac{1}{\pi}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)\right)} \cdot \frac{1}{\mathsf{PI}\left(\right)} \]

       2. +-rgt-identity N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

       7. PI-lowering-PI.f64 99.7

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

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot \ell\right)} \cdot \frac{1}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 50000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 50000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{l\_m}{F}}{\frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 50000.0)
    (- (* PI l_m) (/ (/ l_m F) (/ F PI)))
    (* (* l_m (* PI PI)) (/ 1.0 PI)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 50000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m / F) / (F / ((double) M_PI)));
	} else {
		tmp = (l_m * (((double) M_PI) * ((double) M_PI))) * (1.0 / ((double) M_PI));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 50000.0) {
		tmp = (Math.PI * l_m) - ((l_m / F) / (F / Math.PI));
	} else {
		tmp = (l_m * (Math.PI * Math.PI)) * (1.0 / Math.PI);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 50000.0:
		tmp = (math.pi * l_m) - ((l_m / F) / (F / math.pi))
	else:
		tmp = (l_m * (math.pi * math.pi)) * (1.0 / math.pi)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 50000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) / Float64(F / pi)));
	else
		tmp = Float64(Float64(l_m * Float64(pi * pi)) * Float64(1.0 / pi));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 50000.0)
		tmp = (pi * l_m) - ((l_m / F) / (F / pi));
	else
		tmp = (l_m * (pi * pi)) * (1.0 / pi);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 50000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 80.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 91.1

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in l around 0

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

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

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

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

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

      3. PI-lowering-PI.f64 84.0

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

   7. Simplified 84.0%

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

      1. associate-/l* N/A

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

      2. associate-*l/ N/A

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

      3. un-div-inv N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. PI-lowering-PI.f64 84.0

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

   9. Applied egg-rr 84.0%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.6

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. *-rgt-identity N/A

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

      2. rgt-mult-inverse N/A

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

      3. clear-num N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)} \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \frac{\ell}{\mathsf{PI}\left(\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \ell}{\mathsf{PI}\left(\right)}} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\ell}\right)} \cdot \ell}{\mathsf{PI}\left(\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{\ell} \cdot \ell\right)}}{\mathsf{PI}\left(\right)} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \mathsf{fma}\left(\pi, \ell, 0\right), 0\right) \cdot \frac{1}{\pi}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)\right)} \cdot \frac{1}{\mathsf{PI}\left(\right)} \]

       2. +-rgt-identity N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

       7. PI-lowering-PI.f64 99.7

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

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot \ell\right)} \cdot \frac{1}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 50000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{F}}{\frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 50000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 50000.0)
    (- (* PI l_m) (/ (* PI (/ l_m F)) F))
    (* (* l_m (* PI PI)) (/ 1.0 PI)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 50000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else {
		tmp = (l_m * (((double) M_PI) * ((double) M_PI))) * (1.0 / ((double) M_PI));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 50000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else {
		tmp = (l_m * (Math.PI * Math.PI)) * (1.0 / Math.PI);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 50000.0:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	else:
		tmp = (l_m * (math.pi * math.pi)) * (1.0 / math.pi)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 50000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(Float64(l_m * Float64(pi * pi)) * Float64(1.0 / pi));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 50000.0)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	else
		tmp = (l_m * (pi * pi)) * (1.0 / pi);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 50000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 80.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 91.1

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in l around 0

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

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

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

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

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

      3. PI-lowering-PI.f64 84.0

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

   7. Simplified 84.0%

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

      1. associate-*l/ N/A

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

      2. un-div-inv N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. un-div-inv N/A

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

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

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

      8. PI-lowering-PI.f64 84.0

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

   9. Applied egg-rr 84.0%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.6

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. *-rgt-identity N/A

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

      2. rgt-mult-inverse N/A

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

      3. clear-num N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)} \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \frac{\ell}{\mathsf{PI}\left(\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \ell}{\mathsf{PI}\left(\right)}} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\ell}\right)} \cdot \ell}{\mathsf{PI}\left(\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{\ell} \cdot \ell\right)}}{\mathsf{PI}\left(\right)} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \mathsf{fma}\left(\pi, \ell, 0\right), 0\right) \cdot \frac{1}{\pi}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)\right)} \cdot \frac{1}{\mathsf{PI}\left(\right)} \]

       2. +-rgt-identity N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

       7. PI-lowering-PI.f64 99.7

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

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot \ell\right)} \cdot \frac{1}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 50000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 50000:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 50000.0)
    (* PI (- l_m (/ l_m (* F F))))
    (* (* l_m (* PI PI)) (/ 1.0 PI)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 50000.0) {
		tmp = ((double) M_PI) * (l_m - (l_m / (F * F)));
	} else {
		tmp = (l_m * (((double) M_PI) * ((double) M_PI))) * (1.0 / ((double) M_PI));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 50000.0) {
		tmp = Math.PI * (l_m - (l_m / (F * F)));
	} else {
		tmp = (l_m * (Math.PI * Math.PI)) * (1.0 / Math.PI);
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 50000.0:
		tmp = math.pi * (l_m - (l_m / (F * F)))
	else:
		tmp = (l_m * (math.pi * math.pi)) * (1.0 / math.pi)
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 50000.0)
		tmp = Float64(pi * Float64(l_m - Float64(l_m / Float64(F * F))));
	else
		tmp = Float64(Float64(l_m * Float64(pi * pi)) * Float64(1.0 / pi));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 50000.0)
		tmp = pi * (l_m - (l_m / (F * F)));
	else
		tmp = (l_m * (pi * pi)) * (1.0 / pi);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 50000.0], N[(Pi * N[(l$95$m - N[(l$95$m / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 80.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. un-div-inv N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-rgt-identity N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac2 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

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

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.8

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

   7. Simplified 74.8%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.6

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. *-rgt-identity N/A

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

      2. rgt-mult-inverse N/A

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

      3. clear-num N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)} \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \frac{\ell}{\mathsf{PI}\left(\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}{\ell} \cdot \ell}{\mathsf{PI}\left(\right)}} \]

      8. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\ell}\right)} \cdot \ell}{\mathsf{PI}\left(\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{\ell} \cdot \ell\right)}}{\mathsf{PI}\left(\right)} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \]

      12. div-inv N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

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

          \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \ell + 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, \mathsf{fma}\left(\pi, \ell, 0\right), 0\right) \cdot \frac{1}{\pi}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \ell + 0\right)\right)} \cdot \frac{1}{\mathsf{PI}\left(\right)} \]

       2. +-rgt-identity N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

       7. PI-lowering-PI.f64 99.7

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

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot \ell\right)} \cdot \frac{1}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 50000:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{1}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 50000:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 50000.0) (* PI (- l_m (/ 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) <= 50000.0) {
		tmp = ((double) M_PI) * (l_m - (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) <= 50000.0) {
		tmp = Math.PI * (l_m - (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) <= 50000.0:
		tmp = math.pi * (l_m - (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) <= 50000.0)
		tmp = Float64(pi * Float64(l_m - Float64(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) <= 50000.0)
		tmp = pi * (l_m - (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], 50000.0], N[(Pi * N[(l$95$m - N[(l$95$m / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 80.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. un-div-inv N/A

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

      5. distribute-neg-frac2 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-rgt-identity N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac2 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

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

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.8

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

   7. Simplified 74.8%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.6

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 99.6

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

   7. Applied egg-rr 99.6%

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

Alternative 6: 92.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 50000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= (* PI l_m) 50000.0) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 50000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 50000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 50000.0:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 50000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 50000.0)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 50000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 73.8

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

   5. Simplified 73.8%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 57.3%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.6

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. PI-lowering-PI.f64 99.6

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

   7. Applied egg-rr 99.6%

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

Alternative 7: 74.8% accurate, 22.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

   3. PI-lowering-PI.f64 76.7

      \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

5. Simplified 76.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

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

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

   4. PI-lowering-PI.f64 76.7

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

7. Applied egg-rr 76.7%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))