VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 99.0%

Time: 16.8s

Alternatives: 11

Speedup: 3.7×

• could not determine a ground truth

  1. F = 2.887358321390972e+255
  2. l = 2641755806446210.5

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: 76.6% 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: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 40000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 40000000.0)
    (- (* 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) <= 40000000.0) {
		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) <= 40000000.0) {
		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) <= 40000000.0:
		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) <= 40000000.0)
		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) <= 40000000.0)
		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], 40000000.0], 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) < 4e7

   1. Initial program 77.4%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 86.9

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

   4. Applied egg-rr 86.9%

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

   if 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 72.1%

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

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

      2. PI-lowering-PI.f64 99.6

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

   5. Simplified 99.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 40000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\pi \cdot \frac{l\_m}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -2e-208)
    (* 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) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)))) <= -2e-208) {
		tmp = ((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) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)))) <= -2e-208) {
		tmp = 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) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))) <= -2e-208:
		tmp = 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(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F)))) <= -2e-208)
		tmp = Float64(pi * Float64(l_m / Float64(F * Float64(-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) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -2e-208)
		tmp = 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[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-208], N[(Pi * N[(l$95$m / N[(F * (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -2.0000000000000002e-208

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 63.9

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

   5. Simplified 63.9%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 19.6

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

   8. Simplified 19.6%

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

      1. *-commutative N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 20.2

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

   10. Applied egg-rr 20.2%

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

   if -2.0000000000000002e-208 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.9%

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

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

      2. PI-lowering-PI.f64 81.1

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

   5. Simplified 81.1%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\pi \cdot \frac{\ell}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;l\_m \cdot \frac{\pi}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F)))) -2e-208)
    (* l_m (/ PI (* F (- F))))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (((((double) M_PI) * l_m) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)))) <= -2e-208) {
		tmp = l_m * (((double) M_PI) / (F * -F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (((Math.PI * l_m) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)))) <= -2e-208) {
		tmp = l_m * (Math.PI / (F * -F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if ((math.pi * l_m) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))) <= -2e-208:
		tmp = l_m * (math.pi / (F * -F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F)))) <= -2e-208)
		tmp = Float64(l_m * Float64(pi / Float64(F * Float64(-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) + (tan((pi * l_m)) * (-1.0 / (F * F)))) <= -2e-208)
		tmp = l_m * (pi / (F * -F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-208], N[(l$95$m * N[(Pi / N[(F * (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -2.0000000000000002e-208

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 63.9

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

   5. Simplified 63.9%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 19.6

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

   8. Simplified 19.6%

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

   if -2.0000000000000002e-208 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.9%

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

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

      2. PI-lowering-PI.f64 81.1

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

   5. Simplified 81.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\ell \cdot \frac{\pi}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(l\_m, l\_m \cdot \mathsf{fma}\left(l\_m, l\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot 0.13333333333333333\right), \pi \cdot \left(\pi \cdot \left(\pi \cdot 0.3333333333333333\right)\right)\right), \pi\right)}{-F}, \frac{l\_m}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000.0)
    (fma
     (/
      (fma
       l_m
       (*
        l_m
        (fma
         l_m
         (* l_m (* (* PI (* PI (* PI (* PI PI)))) 0.13333333333333333))
         (* PI (* PI (* PI 0.3333333333333333)))))
       PI)
      (- F))
     (/ l_m F)
     (* PI l_m))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000.0) {
		tmp = fma((fma(l_m, (l_m * fma(l_m, (l_m * ((((double) M_PI) * (((double) M_PI) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))) * 0.13333333333333333)), (((double) M_PI) * (((double) M_PI) * (((double) M_PI) * 0.3333333333333333))))), ((double) M_PI)) / -F), (l_m / F), (((double) M_PI) * l_m));
	} else {
		tmp = ((double) M_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) <= 5000.0)
		tmp = fma(Float64(fma(l_m, Float64(l_m * fma(l_m, Float64(l_m * Float64(Float64(pi * Float64(pi * Float64(pi * Float64(pi * pi)))) * 0.13333333333333333)), Float64(pi * Float64(pi * Float64(pi * 0.3333333333333333))))), pi) / Float64(-F)), Float64(l_m / F), Float64(pi * l_m));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(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], 5000.0], N[(N[(N[(l$95$m * N[(l$95$m * N[(l$95$m * N[(l$95$m * N[(N[(Pi * N[(Pi * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(Pi * N[(Pi * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + Pi), $MachinePrecision] / (-F)), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 77.5%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 61.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\pi, \left(\pi \cdot \pi\right) \cdot 0.3333333333333333, \left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), -0.03333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.16666666666666666\right)\right)\right), \pi\right)\right)} \]

   7. Applied egg-rr 71.2%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\ell, \ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot 0.13333333333333333\right), \pi \cdot \left(\pi \cdot \left(\pi \cdot 0.3333333333333333\right)\right)\right), \pi\right)}{-F}, \frac{\ell}{F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;\pi \cdot l\_m + \frac{l\_m \cdot \mathsf{fma}\left(\left(l\_m \cdot l\_m\right) \cdot \frac{\pi \cdot \left(\pi \cdot \pi\right)}{F}, -0.3333333333333333, \frac{-\pi}{F}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000.0)
    (+
     (* PI l_m)
     (/
      (*
       l_m
       (fma
        (* (* l_m l_m) (/ (* PI (* PI PI)) F))
        -0.3333333333333333
        (/ (- 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) <= 5000.0) {
		tmp = (((double) M_PI) * l_m) + ((l_m * fma(((l_m * l_m) * ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) / F)), -0.3333333333333333, (-((double) M_PI) / F))) / F);
	} else {
		tmp = ((double) M_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) <= 5000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(l_m * fma(Float64(Float64(l_m * l_m) * Float64(Float64(pi * Float64(pi * pi)) / F)), -0.3333333333333333, Float64(Float64(-pi) / F))) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(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], 5000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(l$95$m * N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 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) < 5e3

   1. Initial program 77.5%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      12. neg-lowering-neg.f64 87.1

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

   4. Applied egg-rr 87.1%

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

   5. Taylor expanded in l around 0

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

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

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\ell \cdot \left(-1 \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)\right) + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}}{\mathsf{neg}\left(F\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \left(\color{blue}{\left(\mathsf{neg}\left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)\right)\right)} + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}{\mathsf{neg}\left(F\right)} \]

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

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \left(\left(\mathsf{neg}\left({\ell}^{2} \cdot \color{blue}{\left(\frac{{\mathsf{PI}\left(\right)}^{3}}{F} \cdot \left(\frac{-1}{6} - \frac{-1}{2}\right)\right)}\right)\right) + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}{\mathsf{neg}\left(F\right)} \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \left(\left(\mathsf{neg}\left({\ell}^{2} \cdot \left(\frac{{\mathsf{PI}\left(\right)}^{3}}{F} \cdot \color{blue}{\frac{1}{3}}\right)\right)\right) + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}{\mathsf{neg}\left(F\right)} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right) \cdot \frac{1}{3}}\right)\right) + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}{\mathsf{neg}\left(F\right)} \]

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

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}{\mathsf{neg}\left(F\right)} \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \left(\left({\ell}^{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right) \cdot \color{blue}{\frac{-1}{3}} + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}{\mathsf{neg}\left(F\right)} \]

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

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}, \frac{-1}{3}, -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}\right)}}{\mathsf{neg}\left(F\right)} \]

   7. Simplified 73.1%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\pi \cdot \ell + \frac{\ell \cdot \mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{\pi \cdot \left(\pi \cdot \pi\right)}{F}, -0.3333333333333333, \frac{-\pi}{F}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\left(l\_m \cdot l\_m\right) \cdot \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.3333333333333333}{F} - \frac{\pi}{F}, \frac{l\_m}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000.0)
    (fma
     (-
      (* (* l_m l_m) (/ (* (* PI (* PI PI)) -0.3333333333333333) F))
      (/ PI F))
     (/ l_m F)
     (* PI l_m))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000.0) {
		tmp = fma((((l_m * l_m) * (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -0.3333333333333333) / F)) - (((double) M_PI) / F)), (l_m / F), (((double) M_PI) * l_m));
	} else {
		tmp = ((double) M_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) <= 5000.0)
		tmp = fma(Float64(Float64(Float64(l_m * l_m) * Float64(Float64(Float64(pi * Float64(pi * pi)) * -0.3333333333333333) / F)) - Float64(pi / F)), Float64(l_m / F), Float64(pi * l_m));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(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], 5000.0], N[(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] * N[(l$95$m / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 77.5%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 61.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\pi, \left(\pi \cdot \pi\right) \cdot 0.3333333333333333, \left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), -0.03333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.16666666666666666\right)\right)\right), \pi\right)\right)} \]

   7. Applied egg-rr 71.2%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{F} + -1 \cdot \frac{\mathsf{PI}\left(\right)}{F}}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \frac{{\ell}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{F} + \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{F}\right)\right)}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{F} - \frac{\mathsf{PI}\left(\right)}{F}}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\ell}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{F} \cdot \frac{-1}{3}} - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\ell}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{F} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{\ell}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{F} \cdot \frac{1}{3}\right)\right)} - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left({\ell}^{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)} \cdot \frac{1}{3}\right)\right) - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{{\mathsf{PI}\left(\right)}^{3}}{F} \cdot \frac{1}{3}\right)}\right)\right) - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left({\ell}^{2} \cdot \left(\frac{{\mathsf{PI}\left(\right)}^{3}}{F} \cdot \color{blue}{\left(\frac{-1}{6} - \frac{-1}{2}\right)}\right)\right)\right) - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

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

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left({\ell}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)}\right)\right) - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)\right)} - \frac{\mathsf{PI}\left(\right)}{F}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \left({\ell}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F} - \frac{-1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{F}\right)\right) - \frac{\mathsf{PI}\left(\right)}{F}}, \frac{\ell}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]

   10. Simplified 73.1%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.3333333333333333}{F} - \frac{\pi}{F}, \frac{\ell}{F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5000.0)
    (- (* PI l_m) (/ (* l_m (/ PI F)) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 5000.0) {
		tmp = (Math.PI * l_m) - ((l_m * (Math.PI / F)) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 5000.0:
		tmp = (math.pi * l_m) - ((l_m * (math.pi / F)) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 5000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 5000.0)
		tmp = (pi * l_m) - ((l_m * (pi / F)) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 77.5%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      12. neg-lowering-neg.f64 87.1

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

   4. Applied egg-rr 87.1%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. PI-lowering-PI.f64 79.5

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

   7. Simplified 79.5%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\pi}{F}, \frac{l\_m}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000.0)
    (fma (/ (- PI) F) (/ l_m F) (* PI l_m))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000.0) {
		tmp = fma((-((double) M_PI) / F), (l_m / F), (((double) M_PI) * l_m));
	} else {
		tmp = ((double) M_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) <= 5000.0)
		tmp = fma(Float64(Float64(-pi) / F), Float64(l_m / F), Float64(pi * l_m));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(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], 5000.0], N[(N[((-Pi) / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 77.5%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 61.6%

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\ell \cdot \mathsf{fma}\left(\ell \cdot \ell, \mathsf{fma}\left(\pi, \left(\pi \cdot \pi\right) \cdot 0.3333333333333333, \left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), -0.03333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.16666666666666666\right)\right)\right), \pi\right)\right)} \]

   7. Applied egg-rr 71.2%

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

   8. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

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

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

      5. neg-lowering-neg.f64 79.5

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

   10. Simplified 79.5%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-\pi}{F}, \frac{\ell}{F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5000.0)
    (- (* PI l_m) (/ (* PI l_m) (* F F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * l_m) / (F * F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 5000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI * l_m) / (F * F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 5000.0:
		tmp = (math.pi * l_m) - ((math.pi * l_m) / (F * F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 5000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * l_m) / Float64(F * F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 5000.0)
		tmp = (pi * l_m) - ((pi * l_m) / (F * F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * l$95$m), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 77.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 70.7

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

   5. Simplified 70.7%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= (* PI l_m) 5000.0) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 5000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 5000.0:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 5000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 5000.0)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 77.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 69.9

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

   5. Simplified 69.9%

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

   if 5e3 < (*.f64 (PI.f64) l)

   1. Initial program 71.9%

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

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

      2. PI-lowering-PI.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.8% accurate, 22.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in l around inf

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

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

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

   2. PI-lowering-PI.f64 77.0

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

5. Simplified 77.0%

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

6. Final simplification 77.0%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))