VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.8% → 99.1%

Time: 18.7s

Alternatives: 6

Speedup: 8.1×

• could not determine a ground truth

  1. F = 2.0820789029007703e-100
  2. l = -3064230591156657.5

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: 76.8% 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: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 1000000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{F \cdot \frac{F}{\tan \left(\pi \cdot l\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1000000000000.0)
    (+ (* PI l_m) (/ -1.0 (* F (/ F (tan (* PI l_m))))))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1000000000000.0) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F * (F / tan((((double) M_PI) * l_m)))));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1000000000000.0) {
		tmp = (Math.PI * l_m) + (-1.0 / (F * (F / Math.tan((Math.PI * l_m)))));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1000000000000.0:
		tmp = (math.pi * l_m) + (-1.0 / (F * (F / math.tan((math.pi * l_m)))))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F * Float64(F / tan(Float64(pi * l_m))))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1000000000000.0)
		tmp = (pi * l_m) + (-1.0 / (F * (F / tan((pi * l_m)))));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F * N[(F / N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e12

   1. Initial program 82.8%

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

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. PI-lowering-PI.f64 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 90.1%

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

   6. Applied egg-rr 90.1%

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

   if 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 70.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 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 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 1000000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{F \cdot \frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 1000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 1000000000000.0)
    (- (* 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) <= 1000000000000.0) {
		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) <= 1000000000000.0) {
		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) <= 1000000000000.0:
		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) <= 1000000000000.0)
		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) <= 1000000000000.0)
		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], 1000000000000.0], 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) < 1e12

   1. Initial program 82.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 90.0%

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

   4. Applied egg-rr 90.0%

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

   if 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 70.0%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 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 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 1000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 20000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 20000000.0)
    (- (* PI l_m) (* (/ PI F) (/ l_m F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 20000000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 20000000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 20000000.0:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 20000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 20000000.0)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 20000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e7

   1. Initial program 82.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 77.8%

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

   5. Simplified 77.8%

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 78.3%

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

   7. Applied egg-rr 78.3%

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

      1. times-frac N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 85.1%

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

   9. Applied egg-rr 85.1%

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

   if 2e7 < (*.f64 (PI.f64) l)

   1. Initial program 69.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 20000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 480000000:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 480000000.0) (* PI (- l_m (/ l_m (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 480000000.0) {
		tmp = ((double) M_PI) * (l_m - (l_m / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 480000000.0) {
		tmp = Math.PI * (l_m - (l_m / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 480000000.0:
		tmp = math.pi * (l_m - (l_m / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 480000000.0)
		tmp = Float64(pi * Float64(l_m - Float64(l_m / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 480000000.0)
		tmp = pi * (l_m - (l_m / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 480000000.0], N[(Pi * N[(l$95$m - N[(l$95$m / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.8e8

   1. Initial program 82.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 77.8%

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

   5. Simplified 77.8%

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 78.3%

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

   7. Applied egg-rr 78.3%

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

      1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 78.3%

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

   9. Applied egg-rr 78.3%

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

   if 4.8e8 < l

   1. Initial program 69.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 480000000:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 480000000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 480000000.0) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 480000000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 480000000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 480000000.0:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 480000000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 480000000.0)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 480000000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.8e8

   1. Initial program 82.9%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      10. PI-lowering-PI.f64 90.3%

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   6. 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 77.8%

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

   7. Simplified 77.8%

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

   if 4.8e8 < l

   1. Initial program 69.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 480000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% 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 79.6%

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

3. Taylor expanded in l around inf

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

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

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

   2. PI-lowering-PI.f64 74.2%

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

5. Simplified 74.2%

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

6. Final simplification 74.2%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))