VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 99.2%

Time: 19.4s

Alternatives: 9

Speedup: 7.1×

• could not determine a ground truth

  1. F = -6.532413301353679e-55
  2. l = -3274760115324495.5

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: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 20000000000000.0)
    (+ (* PI l_m) (/ (* (/ -1.0 F) (tan (* PI l_m))) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l_m) + (((-1.0 / F) * tan((((double) M_PI) * l_m))) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 20000000000000.0) {
		tmp = (Math.PI * l_m) + (((-1.0 / F) * Math.tan((Math.PI * l_m))) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 20000000000000.0:
		tmp = (math.pi * l_m) + (((-1.0 / F) * math.tan((math.pi * l_m))) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(-1.0 / F) * tan(Float64(pi * l_m))) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 20000000000000.0)
		tmp = (pi * l_m) + (((-1.0 / F) * tan((pi * l_m))) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(-1.0 / F), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.6%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right)\right), \left(0 - \color{blue}{F}\right)\right)\right) \]

      13. --lowering--.f64 89.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{F}\right)\right)\right) \]

   4. Applied egg-rr 89.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{-1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{0 - F}} \]

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

   1. Initial program 69.5%

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

   3. Taylor expanded in l around inf

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

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

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

      2. PI-lowering-PI.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

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) 20000000000000.0)
    (- (* 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) <= 20000000000000.0) {
		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) <= 20000000000000.0) {
		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) <= 20000000000000.0:
		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) <= 20000000000000.0)
		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) <= 20000000000000.0)
		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], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.6%

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

      1. *-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. un-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 89.0%

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

   4. Applied egg-rr 89.0%

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

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

   1. Initial program 69.5%

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

   3. Taylor expanded in l around inf

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

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

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

      2. PI-lowering-PI.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 91.9%

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

Alternative 3: 98.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(\pi \cdot \pi\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.05:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{l\_m \cdot \left(\pi + \left(l\_m \cdot l\_m\right) \cdot \left(t\_0 \cdot 0.3333333333333333 + \left(l\_m \cdot l\_m\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot t\_0\right)\right) \cdot 0.13333333333333333\right)\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* PI (* PI PI))))
   (*
    l_s
    (if (<= (* PI l_m) 0.05)
      (-
       (* PI l_m)
       (/
        (/
         (*
          l_m
          (+
           PI
           (*
            (* l_m l_m)
            (+
             (* t_0 0.3333333333333333)
             (* (* l_m l_m) (* (* PI (* PI t_0)) 0.13333333333333333))))))
         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 t_0 = ((double) M_PI) * (((double) M_PI) * ((double) M_PI));
	double tmp;
	if ((((double) M_PI) * l_m) <= 0.05) {
		tmp = (((double) M_PI) * l_m) - (((l_m * (((double) M_PI) + ((l_m * l_m) * ((t_0 * 0.3333333333333333) + ((l_m * l_m) * ((((double) M_PI) * (((double) M_PI) * t_0)) * 0.13333333333333333)))))) / 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 t_0 = Math.PI * (Math.PI * Math.PI);
	double tmp;
	if ((Math.PI * l_m) <= 0.05) {
		tmp = (Math.PI * l_m) - (((l_m * (Math.PI + ((l_m * l_m) * ((t_0 * 0.3333333333333333) + ((l_m * l_m) * ((Math.PI * (Math.PI * t_0)) * 0.13333333333333333)))))) / 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):
	t_0 = math.pi * (math.pi * math.pi)
	tmp = 0
	if (math.pi * l_m) <= 0.05:
		tmp = (math.pi * l_m) - (((l_m * (math.pi + ((l_m * l_m) * ((t_0 * 0.3333333333333333) + ((l_m * l_m) * ((math.pi * (math.pi * t_0)) * 0.13333333333333333)))))) / 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)
	t_0 = Float64(pi * Float64(pi * pi))
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.05)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(l_m * Float64(pi + Float64(Float64(l_m * l_m) * Float64(Float64(t_0 * 0.3333333333333333) + Float64(Float64(l_m * l_m) * Float64(Float64(pi * Float64(pi * t_0)) * 0.13333333333333333)))))) / 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)
	t_0 = pi * (pi * pi);
	tmp = 0.0;
	if ((pi * l_m) <= 0.05)
		tmp = (pi * l_m) - (((l_m * (pi + ((l_m * l_m) * ((t_0 * 0.3333333333333333) + ((l_m * l_m) * ((pi * (pi * t_0)) * 0.13333333333333333)))))) / 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_] := Block[{t$95$0 = N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.05], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(l$95$m * N[(Pi + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] + N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(Pi * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) < 0.050000000000000003

   1. Initial program 78.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(F, F\right)\right), \color{blue}{\left(\ell \cdot \left(\mathsf{PI}\left(\right) + {\ell}^{2} \cdot \left(\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {\ell}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} - \left(\frac{-1}{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right) - \frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}\right)\right) \]

   5. Simplified 64.4%

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \left(\pi + \left(\ell \cdot \ell\right) \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.3333333333333333\right) + \left(\ell \cdot \ell\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot -0.03333333333333333 + \left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.16666666666666666\right)\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\left(\ell \cdot \left(\mathsf{PI}\left(\right) + \left(\ell \cdot \ell\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right) + \left(\ell \cdot \ell\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \frac{-1}{30} + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right)\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\ell \cdot \left(\mathsf{PI}\left(\right) + \left(\ell \cdot \ell\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right) + \left(\ell \cdot \ell\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \frac{-1}{30} + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{\color{blue}{F \cdot F}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \left(\frac{\frac{\ell \cdot \left(\mathsf{PI}\left(\right) + \left(\ell \cdot \ell\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right) + \left(\ell \cdot \ell\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \frac{-1}{30} + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{F}}{\color{blue}{F}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\mathsf{PI}\left(\right) + \left(\ell \cdot \ell\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}\right) + \left(\ell \cdot \ell\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \frac{-1}{30} + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{6}\right)\right)\right)\right)}{F}\right), \color{blue}{F}\right)\right) \]

   7. Applied egg-rr 74.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\ell \cdot \left(\pi + \left(\ell \cdot \ell\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.3333333333333333 + \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot 0.13333333333333333\right) \cdot \left(\ell \cdot \ell\right)\right)\right)}{F}}{F}} \]

   if 0.050000000000000003 < (*.f64 (PI.f64) l)

   1. Initial program 69.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.05:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell \cdot \left(\pi + \left(\ell \cdot \ell\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot 0.13333333333333333\right)\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{-1}{F}}{\left(l\_m \cdot \left(\left(\pi \cdot l\_m\right) \cdot -0.3333333333333333\right) + \frac{1}{\pi}\right) \cdot \frac{F}{l\_m}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 20000000000000.0)
    (+
     (* PI l_m)
     (/
      (/ -1.0 F)
      (* (+ (* l_m (* (* PI l_m) -0.3333333333333333)) (/ 1.0 PI)) (/ F l_m))))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l_m) + ((-1.0 / F) / (((l_m * ((((double) M_PI) * l_m) * -0.3333333333333333)) + (1.0 / ((double) M_PI))) * (F / l_m)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 20000000000000.0) {
		tmp = (Math.PI * l_m) + ((-1.0 / F) / (((l_m * ((Math.PI * l_m) * -0.3333333333333333)) + (1.0 / Math.PI)) * (F / l_m)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 20000000000000.0:
		tmp = (math.pi * l_m) + ((-1.0 / F) / (((l_m * ((math.pi * l_m) * -0.3333333333333333)) + (1.0 / math.pi)) * (F / l_m)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(-1.0 / F) / Float64(Float64(Float64(l_m * Float64(Float64(pi * l_m) * -0.3333333333333333)) + Float64(1.0 / pi)) * Float64(F / l_m))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 20000000000000.0)
		tmp = (pi * l_m) + ((-1.0 / F) / (((l_m * ((pi * l_m) * -0.3333333333333333)) + (1.0 / pi)) * (F / l_m)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(N[(N[(l$95$m * N[(N[(Pi * l$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[(F / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.6%

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

      1. *-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. clear-num N/A

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

      4. associate-/r* N/A

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

      5. frac-times N/A

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

      6. div-inv N/A

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

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

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

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

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

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

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

      10. clear-num N/A

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

      11. tan-quot N/A

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

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

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

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

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

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

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

      15. PI-lowering-PI.f64 89.0%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in l around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 93.9%

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

   7. Simplified 93.9%

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

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

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 N/A

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

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

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

      16. /-lowering-/.f64 88.1%

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

   9. Applied egg-rr 88.1%

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

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

   1. Initial program 69.5%

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

   3. Taylor expanded in l around inf

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

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

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

      2. PI-lowering-PI.f64 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\left(\ell \cdot \left(\left(\pi \cdot \ell\right) \cdot -0.3333333333333333\right) + \frac{1}{\pi}\right) \cdot \frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \frac{\frac{\pi \cdot l\_m}{F}}{0 - F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 7.2 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-74}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (/ (/ (* PI l_m) F) (- 0.0 F))))
   (*
    l_s
    (if (<= F 7.2e-166)
      t_0
      (if (<= F 4.9e-74) (* PI l_m) (if (<= F 8.8e-19) t_0 (* PI l_m)))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = ((((double) M_PI) * l_m) / F) / (0.0 - F);
	double tmp;
	if (F <= 7.2e-166) {
		tmp = t_0;
	} else if (F <= 4.9e-74) {
		tmp = ((double) M_PI) * l_m;
	} else if (F <= 8.8e-19) {
		tmp = t_0;
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double t_0 = ((Math.PI * l_m) / F) / (0.0 - F);
	double tmp;
	if (F <= 7.2e-166) {
		tmp = t_0;
	} else if (F <= 4.9e-74) {
		tmp = Math.PI * l_m;
	} else if (F <= 8.8e-19) {
		tmp = t_0;
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = ((math.pi * l_m) / F) / (0.0 - F)
	tmp = 0
	if F <= 7.2e-166:
		tmp = t_0
	elif F <= 4.9e-74:
		tmp = math.pi * l_m
	elif F <= 8.8e-19:
		tmp = t_0
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(Float64(Float64(pi * l_m) / F) / Float64(0.0 - F))
	tmp = 0.0
	if (F <= 7.2e-166)
		tmp = t_0;
	elseif (F <= 4.9e-74)
		tmp = Float64(pi * l_m);
	elseif (F <= 8.8e-19)
		tmp = t_0;
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	t_0 = ((pi * l_m) / F) / (0.0 - F);
	tmp = 0.0;
	if (F <= 7.2e-166)
		tmp = t_0;
	elseif (F <= 4.9e-74)
		tmp = pi * l_m;
	elseif (F <= 8.8e-19)
		tmp = t_0;
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / N[(0.0 - F), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[F, 7.2e-166], t$95$0, If[LessEqual[F, 4.9e-74], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[F, 8.8e-19], t$95$0, N[(Pi * l$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 7.2000000000000002e-166 or 4.9000000000000003e-74 < F < 8.7999999999999994e-19

   1. Initial program 67.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   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 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)}\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{\color{blue}{{F}^{2}}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{F \cdot \color{blue}{F}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}{\color{blue}{F}}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}\right), \color{blue}{F}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}\right), F\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \mathsf{PI}\left(\right)\right), F\right), F\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI}\left(\right)\right), F\right), F\right)\right) \]

      13. PI-lowering-PI.f64 38.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right), F\right), F\right)\right) \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{0 - \frac{\frac{\ell \cdot \pi}{F}}{F}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. PI-lowering-PI.f64 38.3%

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

   10. Applied egg-rr 38.3%

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

   if 7.2000000000000002e-166 < F < 4.9000000000000003e-74 or 8.7999999999999994e-19 < F

   1. Initial program 90.7%

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

   3. Taylor expanded in l around 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 90.5%

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

   5. Simplified 90.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{\pi \cdot \ell}{F}}{0 - F}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-74}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\pi \cdot \ell}{F}}{0 - F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.05:\\ \;\;\;\;\pi \cdot l\_m + \frac{\left(\pi \cdot l\_m\right) \cdot \frac{-1}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 0.05)
    (+ (* PI l_m) (/ (* (* PI l_m) (/ -1.0 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 ((((double) M_PI) * l_m) <= 0.05) {
		tmp = (((double) M_PI) * l_m) + (((((double) M_PI) * l_m) * (-1.0 / 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 ((Math.PI * l_m) <= 0.05) {
		tmp = (Math.PI * l_m) + (((Math.PI * l_m) * (-1.0 / 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 (math.pi * l_m) <= 0.05:
		tmp = (math.pi * l_m) + (((math.pi * l_m) * (-1.0 / 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(pi * l_m) <= 0.05)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(pi * l_m) * Float64(-1.0 / 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) <= 0.05)
		tmp = (pi * l_m) + (((pi * l_m) * (-1.0 / 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[(Pi * l$95$m), $MachinePrecision], 0.05], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(Pi * l$95$m), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 0.050000000000000003

   1. Initial program 78.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right)\right), \left(0 - \color{blue}{F}\right)\right)\right) \]

      13. --lowering--.f64 89.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \mathsf{tan.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{F}\right)\right)\right) \]

   4. Applied egg-rr 89.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{-1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{0 - F}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \color{blue}{\left(\ell \cdot \mathsf{PI}\left(\right)\right)}\right), \mathsf{\_.f64}\left(0, F\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \mathsf{*.f64}\left(\ell, \mathsf{PI}\left(\right)\right)\right), \mathsf{\_.f64}\left(0, F\right)\right)\right) \]

      2. PI-lowering-PI.f64 85.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, F\right), \mathsf{*.f64}\left(\ell, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{\_.f64}\left(0, F\right)\right)\right) \]

   7. Simplified 85.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{-1}{F} \cdot \color{blue}{\left(\ell \cdot \pi\right)}}{0 - F} \]

   if 0.050000000000000003 < (*.f64 (PI.f64) l)

   1. Initial program 69.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.05:\\ \;\;\;\;\pi \cdot \ell + \frac{\left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.05:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\pi \cdot l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 0.05) (- (* PI l_m) (/ (/ (* PI l_m) F) F)) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 0.05) {
		tmp = (((double) M_PI) * l_m) - (((((double) M_PI) * l_m) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 0.05) {
		tmp = (Math.PI * l_m) - (((Math.PI * l_m) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 0.05:
		tmp = (math.pi * l_m) - (((math.pi * l_m) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.05)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(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) <= 0.05)
		tmp = (pi * l_m) - (((pi * l_m) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.05], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $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) < 0.050000000000000003

   1. Initial program 78.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. un-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 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in l around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. associate-/l* N/A

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

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

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

      5. associate-/l* N/A

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

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

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

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

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

      8. PI-lowering-PI.f64 85.9%

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

   7. Simplified 85.9%

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

   if 0.050000000000000003 < (*.f64 (PI.f64) l)

   1. Initial program 69.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.05:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.05:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 0.05) (* 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) <= 0.05) {
		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) <= 0.05) {
		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) <= 0.05:
		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) <= 0.05)
		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) <= 0.05)
		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], 0.05], 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) < 0.050000000000000003

   1. Initial program 78.7%

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

   3. Taylor expanded in l around 0

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

      1. *-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 75.4%

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

   5. Simplified 75.4%

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

   if 0.050000000000000003 < (*.f64 (PI.f64) l)

   1. Initial program 69.2%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.05:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% 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 76.1%

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

3. Taylor expanded in l around 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 72.7%

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

5. Simplified 72.7%

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

6. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))