VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 99.2%

Time: 17.2s

Alternatives: 10

Speedup: 3.7×

• could not determine a ground truth

  1. F = 2.756606505174376e-204
  2. l = 37540644420337.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 2 \cdot 10^{+14}:\\ \;\;\;\;l\_m \cdot \pi - \frac{\frac{\tan \left(l\_m \cdot \pi\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 2e+14)
    (- (* l_m PI) (/ (/ (tan (* l_m PI)) F) F))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 2e+14) {
		tmp = (l_m * ((double) M_PI)) - ((tan((l_m * ((double) M_PI))) / F) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * Math.PI) <= 2e+14) {
		tmp = (l_m * Math.PI) - ((Math.tan((l_m * Math.PI)) / F) / F);
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m * math.pi) <= 2e+14:
		tmp = (l_m * math.pi) - ((math.tan((l_m * math.pi)) / F) / F)
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 2e+14)
		tmp = Float64(Float64(l_m * pi) - Float64(Float64(tan(Float64(l_m * pi)) / F) / F));
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m * pi) <= 2e+14)
		tmp = (l_m * pi) - ((tan((l_m * pi)) / F) / F);
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(l$95$m * Pi), $MachinePrecision], 2e+14], N[(N[(l$95$m * Pi), $MachinePrecision] - N[(N[(N[Tan[N[(l$95$m * Pi), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 79.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 84.7

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

   4. Applied rewrites 84.7%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 56.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\tan \left(\ell \cdot \pi\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \pi - \tan \left(l\_m \cdot \pi\right) \cdot \frac{1}{F \cdot F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+261}:\\ \;\;\;\;l\_m \cdot \pi\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-233}:\\ \;\;\;\;\frac{\pi}{\left(-F\right) \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (- (* l_m PI) (* (tan (* l_m PI)) (/ 1.0 (* F F))))))
   (*
    l_s
    (if (<= t_0 -1e+261)
      (* l_m PI)
      (if (<= t_0 -2e-233) (* (/ PI (* (- F) F)) l_m) (* l_m PI))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = (l_m * ((double) M_PI)) - (tan((l_m * ((double) M_PI))) * (1.0 / (F * F)));
	double tmp;
	if (t_0 <= -1e+261) {
		tmp = l_m * ((double) M_PI);
	} else if (t_0 <= -2e-233) {
		tmp = (((double) M_PI) / (-F * F)) * l_m;
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double t_0 = (l_m * Math.PI) - (Math.tan((l_m * Math.PI)) * (1.0 / (F * F)));
	double tmp;
	if (t_0 <= -1e+261) {
		tmp = l_m * Math.PI;
	} else if (t_0 <= -2e-233) {
		tmp = (Math.PI / (-F * F)) * l_m;
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = (l_m * math.pi) - (math.tan((l_m * math.pi)) * (1.0 / (F * F)))
	tmp = 0
	if t_0 <= -1e+261:
		tmp = l_m * math.pi
	elif t_0 <= -2e-233:
		tmp = (math.pi / (-F * F)) * l_m
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(Float64(l_m * pi) - Float64(tan(Float64(l_m * pi)) * Float64(1.0 / Float64(F * F))))
	tmp = 0.0
	if (t_0 <= -1e+261)
		tmp = Float64(l_m * pi);
	elseif (t_0 <= -2e-233)
		tmp = Float64(Float64(pi / Float64(Float64(-F) * F)) * l_m);
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	t_0 = (l_m * pi) - (tan((l_m * pi)) * (1.0 / (F * F)));
	tmp = 0.0;
	if (t_0 <= -1e+261)
		tmp = l_m * pi;
	elseif (t_0 <= -2e-233)
		tmp = (pi / (-F * F)) * l_m;
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := Block[{t$95$0 = N[(N[(l$95$m * Pi), $MachinePrecision] - N[(N[Tan[N[(l$95$m * Pi), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[t$95$0, -1e+261], N[(l$95$m * Pi), $MachinePrecision], If[LessEqual[t$95$0, -2e-233], N[(N[(Pi / N[((-F) * F), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(l$95$m * Pi), $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)))) < -9.9999999999999993e260 or -1.99999999999999992e-233 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 67.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -9.9999999999999993e260 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.99999999999999992e-233

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 91.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.2

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

   4. Applied rewrites 91.2%

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

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.6

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 20.6%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi - \tan \left(\ell \cdot \pi\right) \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{+261}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\ell \cdot \pi - \tan \left(\ell \cdot \pi\right) \cdot \frac{1}{F \cdot F} \leq -2 \cdot 10^{-233}:\\ \;\;\;\;\frac{\pi}{\left(-F\right) \cdot F} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 2 \cdot 10^{+14}:\\ \;\;\;\;l\_m \cdot \pi - \frac{\frac{1}{F}}{\left(\frac{1}{l\_m} \cdot \mathsf{fma}\left(\left(-0.3333333333333333 \cdot l\_m\right) \cdot \pi, l\_m, \frac{1}{\pi}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 2e+14)
    (-
     (* l_m PI)
     (/
      (/ 1.0 F)
      (*
       (* (/ 1.0 l_m) (fma (* (* -0.3333333333333333 l_m) PI) l_m (/ 1.0 PI)))
       F)))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 2e+14) {
		tmp = (l_m * ((double) M_PI)) - ((1.0 / F) / (((1.0 / l_m) * fma(((-0.3333333333333333 * l_m) * ((double) M_PI)), l_m, (1.0 / ((double) M_PI)))) * F));
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 2e+14)
		tmp = Float64(Float64(l_m * pi) - Float64(Float64(1.0 / F) / Float64(Float64(Float64(1.0 / l_m) * fma(Float64(Float64(-0.3333333333333333 * l_m) * pi), l_m, Float64(1.0 / pi))) * F)));
	else
		tmp = Float64(l_m * pi);
	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[(l$95$m * Pi), $MachinePrecision], 2e+14], N[(N[(l$95$m * Pi), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] / N[(N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * l$95$m), $MachinePrecision] * Pi), $MachinePrecision] * l$95$m + N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 79.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-PI.f64 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 89.8

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

   10. Applied rewrites 89.8%

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

   12. Applied rewrites 89.8%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 56.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{1}{F}}{\left(\frac{1}{\ell} \cdot \mathsf{fma}\left(\left(-0.3333333333333333 \cdot \ell\right) \cdot \pi, \ell, \frac{1}{\pi}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot l\_m\right) \cdot \pi, l\_m, \frac{1}{\pi}\right)}{l\_m} \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 2e+14)
    (fma
     PI
     l_m
     (/
      (/ -1.0 F)
      (* (/ (fma (* (* -0.3333333333333333 l_m) PI) l_m (/ 1.0 PI)) l_m) F)))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 2e+14) {
		tmp = fma(((double) M_PI), l_m, ((-1.0 / F) / ((fma(((-0.3333333333333333 * l_m) * ((double) M_PI)), l_m, (1.0 / ((double) M_PI))) / l_m) * F)));
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 2e+14)
		tmp = fma(pi, l_m, Float64(Float64(-1.0 / F) / Float64(Float64(fma(Float64(Float64(-0.3333333333333333 * l_m) * pi), l_m, Float64(1.0 / pi)) / l_m) * F)));
	else
		tmp = Float64(l_m * pi);
	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[(l$95$m * Pi), $MachinePrecision], 2e+14], N[(Pi * l$95$m + N[(N[(-1.0 / F), $MachinePrecision] / N[(N[(N[(N[(N[(-0.3333333333333333 * l$95$m), $MachinePrecision] * Pi), $MachinePrecision] * l$95$m + N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e14

   1. Initial program 79.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-PI.f64 89.8

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

   7. Applied rewrites 89.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

   9. Applied rewrites 83.0%

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

   11. Applied rewrites 89.8%

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

   if 2e14 < (*.f64 (PI.f64) l)

   1. Initial program 56.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \ell\right) \cdot \pi, \ell, \frac{1}{\pi}\right)}{\ell} \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 100000:\\ \;\;\;\;l\_m \cdot \pi - \frac{\frac{l\_m \cdot \pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 100000.0)
    (- (* l_m PI) (/ (/ (* l_m PI) F) F))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 100000.0) {
		tmp = (l_m * ((double) M_PI)) - (((l_m * ((double) M_PI)) / F) / F);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * Math.PI) <= 100000.0) {
		tmp = (l_m * Math.PI) - (((l_m * Math.PI) / F) / F);
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m * math.pi) <= 100000.0:
		tmp = (l_m * math.pi) - (((l_m * math.pi) / F) / F)
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 100000.0)
		tmp = Float64(Float64(l_m * pi) - Float64(Float64(Float64(l_m * pi) / F) / F));
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m * pi) <= 100000.0)
		tmp = (l_m * pi) - (((l_m * pi) / F) / F);
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(l$95$m * Pi), $MachinePrecision], 100000.0], N[(N[(l$95$m * Pi), $MachinePrecision] - N[(N[(N[(l$95$m * Pi), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e5

   1. Initial program 79.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if 1e5 < (*.f64 (PI.f64) l)

   1. Initial program 57.2%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 100000:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\ell \cdot \pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 100000:\\ \;\;\;\;\frac{\left(\left(F - \frac{1}{F}\right) \cdot \pi\right) \cdot l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 100000.0)
    (/ (* (* (- F (/ 1.0 F)) PI) l_m) F)
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 100000.0) {
		tmp = (((F - (1.0 / F)) * ((double) M_PI)) * l_m) / F;
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * Math.PI) <= 100000.0) {
		tmp = (((F - (1.0 / F)) * Math.PI) * l_m) / F;
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m * math.pi) <= 100000.0:
		tmp = (((F - (1.0 / F)) * math.pi) * l_m) / F
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 100000.0)
		tmp = Float64(Float64(Float64(Float64(F - Float64(1.0 / F)) * pi) * l_m) / F);
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m * pi) <= 100000.0)
		tmp = (((F - (1.0 / F)) * pi) * l_m) / F;
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(l$95$m * Pi), $MachinePrecision], 100000.0], N[(N[(N[(N[(F - N[(1.0 / F), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * l$95$m), $MachinePrecision] / F), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e5

   1. Initial program 79.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 84.7

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

   4. Applied rewrites 84.7%

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

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 74.3%

      \[\leadsto \frac{\pi \cdot F - \frac{\pi}{F}}{1 \cdot F} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 72.7%

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

   if 1e5 < (*.f64 (PI.f64) l)

   1. Initial program 57.2%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 100000:\\ \;\;\;\;\frac{\left(\left(F - \frac{1}{F}\right) \cdot \pi\right) \cdot \ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 100000:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{l\_m \cdot \pi}{\left(-F\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 100000.0)
    (fma PI l_m (/ (* l_m PI) (* (- F) F)))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 100000.0) {
		tmp = fma(((double) M_PI), l_m, ((l_m * ((double) M_PI)) / (-F * F)));
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 100000.0)
		tmp = fma(pi, l_m, Float64(Float64(l_m * pi) / Float64(Float64(-F) * F)));
	else
		tmp = Float64(l_m * pi);
	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[(l$95$m * Pi), $MachinePrecision], 100000.0], N[(Pi * l$95$m + N[(N[(l$95$m * Pi), $MachinePrecision] / N[((-F) * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e5

   1. Initial program 79.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-out N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

   7. Applied rewrites 75.6%

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

   if 1e5 < (*.f64 (PI.f64) l)

   1. Initial program 57.2%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 100000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\ell \cdot \pi}{\left(-F\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 100000:\\ \;\;\;\;l\_m \cdot \pi - \frac{\pi}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 100000.0)
    (- (* l_m PI) (* (/ PI (* F F)) l_m))
    (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 100000.0) {
		tmp = (l_m * ((double) M_PI)) - ((((double) M_PI) / (F * F)) * l_m);
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * Math.PI) <= 100000.0) {
		tmp = (l_m * Math.PI) - ((Math.PI / (F * F)) * l_m);
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m * math.pi) <= 100000.0:
		tmp = (l_m * math.pi) - ((math.pi / (F * F)) * l_m)
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 100000.0)
		tmp = Float64(Float64(l_m * pi) - Float64(Float64(pi / Float64(F * F)) * l_m));
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m * pi) <= 100000.0)
		tmp = (l_m * pi) - ((pi / (F * F)) * l_m);
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(l$95$m * Pi), $MachinePrecision], 100000.0], N[(N[(l$95$m * Pi), $MachinePrecision] - N[(N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e5

   1. Initial program 79.5%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if 1e5 < (*.f64 (PI.f64) l)

   1. Initial program 57.2%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 100000:\\ \;\;\;\;\ell \cdot \pi - \frac{\pi}{F \cdot F} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot \pi \leq 100000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \pi\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* l_m PI) 100000.0) (* (- PI (/ PI (* F F))) l_m) (* l_m PI))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * ((double) M_PI)) <= 100000.0) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * l_m;
	} else {
		tmp = l_m * ((double) M_PI);
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m * Math.PI) <= 100000.0) {
		tmp = (Math.PI - (Math.PI / (F * F))) * l_m;
	} else {
		tmp = l_m * Math.PI;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m * math.pi) <= 100000.0:
		tmp = (math.pi - (math.pi / (F * F))) * l_m
	else:
		tmp = l_m * math.pi
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(l_m * pi) <= 100000.0)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * l_m);
	else
		tmp = Float64(l_m * pi);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m * pi) <= 100000.0)
		tmp = (pi - (pi / (F * F))) * l_m;
	else
		tmp = l_m * pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(l$95$m * Pi), $MachinePrecision], 100000.0], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(l$95$m * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e5

   1. Initial program 79.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if 1e5 < (*.f64 (PI.f64) l)

   1. Initial program 57.2%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \pi \leq 100000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 22.5× speedup

Math

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

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 (* l_m PI)))

C

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

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 * (l_m * Math.PI);
}

Python

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

Julia

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

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * (l_m * pi);
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[(l$95$m * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 74.0

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

5. Applied rewrites 74.0%

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

6. Final simplification 74.0%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))