VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.4% → 98.4%

Time: 21.2s

Alternatives: 9

Speedup: 8.1×

• could not determine a ground truth

  1. F = 1.2391592705444157e+115
  2. l = 3179574476971350.5

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: 75.4% 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: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 10000000.0)
    (- (* 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) <= 10000000.0) {
		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) <= 10000000.0) {
		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) <= 10000000.0:
		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) <= 10000000.0)
		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) <= 10000000.0)
		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], 10000000.0], 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) < 1e7

   1. Initial program 79.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 87.2%

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

   4. Applied egg-rr 87.2%

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

   if 1e7 < (*.f64 (PI.f64) l)

   1. Initial program 60.5%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.02:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.3333333333333333}{F} + \frac{\pi}{F}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 0.02)
    (-
     (* PI l_m)
     (/
      (*
       l_m
       (+
        (* (* l_m l_m) (/ (* (* PI (* PI PI)) 0.3333333333333333) F))
        (/ 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) <= 0.02) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((l_m * l_m) * (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.3333333333333333) / F)) + (((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) <= 0.02) {
		tmp = (Math.PI * l_m) - ((l_m * (((l_m * l_m) * (((Math.PI * (Math.PI * Math.PI)) * 0.3333333333333333) / F)) + (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) <= 0.02:
		tmp = (math.pi * l_m) - ((l_m * (((l_m * l_m) * (((math.pi * (math.pi * math.pi)) * 0.3333333333333333) / F)) + (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) <= 0.02)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(Float64(Float64(l_m * l_m) * Float64(Float64(Float64(pi * Float64(pi * pi)) * 0.3333333333333333) / F)) + Float64(pi / F))) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 0.02)
		tmp = (pi * l_m) - ((l_m * (((l_m * l_m) * (((pi * (pi * pi)) * 0.3333333333333333) / F)) + (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], 0.02], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] + N[(Pi / F), $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) < 0.0200000000000000004

   1. Initial program 78.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 87.2%

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in l around 0

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

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

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

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

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

   7. Simplified 73.9%

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

   if 0.0200000000000000004 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.02:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.3333333333333333}{F} + \frac{\pi}{F}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.02:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{l\_m \cdot \left(\pi + l\_m \cdot \left(l\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.3333333333333333\right)\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 0.02)
    (-
     (* PI l_m)
     (/
      (/
       (* l_m (+ PI (* l_m (* l_m (* (* PI (* PI PI)) 0.3333333333333333)))))
       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) <= 0.02) {
		tmp = (((double) M_PI) * l_m) - (((l_m * (((double) M_PI) + (l_m * (l_m * ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.3333333333333333))))) / 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) <= 0.02) {
		tmp = (Math.PI * l_m) - (((l_m * (Math.PI + (l_m * (l_m * ((Math.PI * (Math.PI * Math.PI)) * 0.3333333333333333))))) / 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) <= 0.02:
		tmp = (math.pi * l_m) - (((l_m * (math.pi + (l_m * (l_m * ((math.pi * (math.pi * math.pi)) * 0.3333333333333333))))) / 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) <= 0.02)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(l_m * Float64(pi + Float64(l_m * Float64(l_m * Float64(Float64(pi * Float64(pi * pi)) * 0.3333333333333333))))) / 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) <= 0.02)
		tmp = (pi * l_m) - (((l_m * (pi + (l_m * (l_m * ((pi * (pi * pi)) * 0.3333333333333333))))) / 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], 0.02], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(l$95$m * N[(Pi + N[(l$95$m * N[(l$95$m * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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) < 0.0200000000000000004

   1. Initial program 78.9%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 87.2%

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

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

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

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

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

      18. metadata-eval 72.9%

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

   7. Simplified 72.9%

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

   if 0.0200000000000000004 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.02:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell \cdot \left(\pi + \ell \cdot \left(\ell \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.3333333333333333\right)\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 7.6 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{l\_m}{\frac{F}{\frac{\pi}{0 - F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= F 7.6e-161)
    (* PI l_m)
    (if (<= F 3.6e-96) (/ l_m (/ F (/ PI (- 0.0 F)))) (* PI l_m)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (F <= 7.6e-161) {
		tmp = ((double) M_PI) * l_m;
	} else if (F <= 3.6e-96) {
		tmp = l_m / (F / (((double) M_PI) / (0.0 - F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (F <= 7.6e-161) {
		tmp = Math.PI * l_m;
	} else if (F <= 3.6e-96) {
		tmp = l_m / (F / (Math.PI / (0.0 - F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if F <= 7.6e-161:
		tmp = math.pi * l_m
	elif F <= 3.6e-96:
		tmp = l_m / (F / (math.pi / (0.0 - F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (F <= 7.6e-161)
		tmp = Float64(pi * l_m);
	elseif (F <= 3.6e-96)
		tmp = Float64(l_m / Float64(F / Float64(pi / Float64(0.0 - F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (F <= 7.6e-161)
		tmp = pi * l_m;
	elseif (F <= 3.6e-96)
		tmp = l_m / (F / (pi / (0.0 - F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[F, 7.6e-161], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[F, 3.6e-96], N[(l$95$m / N[(F / N[(Pi / N[(0.0 - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if F < 7.6000000000000003e-161 or 3.60000000000000008e-96 < F

   1. Initial program 75.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 74.0%

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

   5. Simplified 74.0%

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

   if 7.6000000000000003e-161 < F < 3.60000000000000008e-96

   1. Initial program 67.1%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 56.1%

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

   5. Simplified 56.1%

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

   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. neg-sub0 N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. PI-lowering-PI.f64 62.5%

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

   8. Simplified 62.5%

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

      1. sub0-neg N/A

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

      2. clear-num N/A

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

      3. distribute-neg-frac N/A

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

      4. associate-*r/ N/A

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

      5. associate-/r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. associate-/l/ N/A

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

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

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

      9. associate-/l/ N/A

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

      10. associate-*l/ N/A

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

      11. clear-num N/A

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

      12. times-frac N/A

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

      13. associate-/r/ N/A

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

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

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

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

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

      16. /-lowering-/.f64 N/A

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

      17. PI-lowering-PI.f64 58.0%

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

   10. Applied egg-rr 58.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7.6 \cdot 10^{-161}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\ell}{\frac{F}{\frac{\pi}{0 - F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.02:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 0.02) (- (* PI l_m) (* (/ PI F) (/ l_m F))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 0.02) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 0.02) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 0.02:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.02)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 0.02)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.02], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 0.0200000000000000004

   1. Initial program 78.9%

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

   3. Taylor expanded in l around 0

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

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

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

      2. PI-lowering-PI.f64 75.0%

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

   5. Simplified 75.0%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. un-div-inv N/A

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

      4. times-frac N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

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

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

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

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

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

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. /-lowering-/.f64 83.8%

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

   7. Applied egg-rr 83.8%

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

   if 0.0200000000000000004 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.02:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 8.1× speedup

Math

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

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 (<= l_m 0.5) (* 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 (l_m <= 0.5) {
		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 (l_m <= 0.5) {
		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 l_m <= 0.5:
		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 (l_m <= 0.5)
		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 (l_m <= 0.5)
		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[l$95$m, 0.5], 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 l < 0.5

   1. Initial program 78.9%

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

   3. Taylor expanded in l around 0

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

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

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

      2. PI-lowering-PI.f64 75.0%

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

   5. Simplified 75.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 76.7%

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

   7. Applied egg-rr 76.7%

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

   if 0.5 < l

   1. Initial program 61.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 8.1× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 0.5) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 0.5) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 0.5:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 0.5)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 0.5)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 0.5], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 78.9%

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

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

      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 75.5%

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

   5. Simplified 75.5%

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

   if 0.5 < l

   1. Initial program 61.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.5:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{l\_m}{0 - F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 3.7e-22) (* (/ PI F) (/ l_m (- 0.0 F))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 3.7e-22) {
		tmp = (((double) M_PI) / F) * (l_m / (0.0 - F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 3.7e-22) {
		tmp = (Math.PI / F) * (l_m / (0.0 - F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 3.7e-22:
		tmp = (math.pi / F) * (l_m / (0.0 - F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 3.7e-22)
		tmp = Float64(Float64(pi / F) * Float64(l_m / Float64(0.0 - F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 3.7e-22)
		tmp = (pi / F) * (l_m / (0.0 - F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 3.7e-22], N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / N[(0.0 - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.7e-22

   1. Initial program 78.4%

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

   3. Taylor expanded in l around 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 75.2%

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

   5. Simplified 75.2%

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

   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. neg-sub0 N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. PI-lowering-PI.f64 36.9%

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

   8. Simplified 36.9%

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

      1. sub0-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 36.9%

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

   10. Applied egg-rr 36.9%

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

   if 3.7e-22 < l

   1. Initial program 63.9%

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

   3. Taylor expanded in l around inf

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

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

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

      2. PI-lowering-PI.f64 93.8%

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

   5. Simplified 93.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{\ell}{0 - F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
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 74.5%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
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 71.7%

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

5. Simplified 71.7%

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

6. Final simplification 71.7%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))