VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.8% → 99.3%

Time: 19.1s

Alternatives: 11

Speedup: 3.7×

• could not determine a ground truth

  1. F = -1.1711152562492115e+273
  2. l = -2953727280661243.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 85.1

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

   4. Applied egg-rr 85.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.1

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

   6. Applied egg-rr 85.1%

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

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

   1. Initial program 68.0%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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 88.5%

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

Alternative 2: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-265}:\\ \;\;\;\;\frac{\pi \cdot l\_m}{-F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F))))))
   (*
    l_s
    (if (<= t_0 -5e+153)
      (* PI l_m)
      (if (<= t_0 -4e-265) (/ (* 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 t_0 = (((double) M_PI) * l_m) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)));
	double tmp;
	if (t_0 <= -5e+153) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -4e-265) {
		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 t_0 = (Math.PI * l_m) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)));
	double tmp;
	if (t_0 <= -5e+153) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -4e-265) {
		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):
	t_0 = (math.pi * l_m) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))
	tmp = 0
	if t_0 <= -5e+153:
		tmp = math.pi * l_m
	elif t_0 <= -4e-265:
		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)
	t_0 = Float64(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F))))
	tmp = 0.0
	if (t_0 <= -5e+153)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -4e-265)
		tmp = Float64(Float64(pi * l_m) / Float64(-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)
	t_0 = (pi * l_m) + (tan((pi * l_m)) * (-1.0 / (F * F)));
	tmp = 0.0;
	if (t_0 <= -5e+153)
		tmp = pi * l_m;
	elseif (t_0 <= -4e-265)
		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_] := Block[{t$95$0 = 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]}, N[(l$95$s * If[LessEqual[t$95$0, -5e+153], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -4e-265], N[(N[(Pi * l$95$m), $MachinePrecision] / (-N[(F * F), $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)))) < -5.00000000000000018e153 or -3.99999999999999994e-265 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 69.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. lower-*.f64 N/A

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

      2. lower-PI.f64 79.3

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

   5. Simplified 79.3%

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

   if -5.00000000000000018e153 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -3.99999999999999994e-265

   1. Initial program 96.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-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. lower-*.f64 92.8

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

   5. Simplified 92.8%

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

   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-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. lower-*.f64 26.4

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

   8. Simplified 26.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-neg-frac2 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 26.4

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

   10. Applied egg-rr 26.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -4 \cdot 10^{-265}:\\ \;\;\;\;\frac{\pi \cdot \ell}{-F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-265}:\\ \;\;\;\;-l\_m \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F))))))
   (*
    l_s
    (if (<= t_0 -5e+153)
      (* PI l_m)
      (if (<= t_0 -4e-265) (- (* 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 t_0 = (((double) M_PI) * l_m) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)));
	double tmp;
	if (t_0 <= -5e+153) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -4e-265) {
		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 t_0 = (Math.PI * l_m) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)));
	double tmp;
	if (t_0 <= -5e+153) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -4e-265) {
		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):
	t_0 = (math.pi * l_m) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))
	tmp = 0
	if t_0 <= -5e+153:
		tmp = math.pi * l_m
	elif t_0 <= -4e-265:
		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)
	t_0 = Float64(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F))))
	tmp = 0.0
	if (t_0 <= -5e+153)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -4e-265)
		tmp = Float64(-Float64(l_m * 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)
	t_0 = (pi * l_m) + (tan((pi * l_m)) * (-1.0 / (F * F)));
	tmp = 0.0;
	if (t_0 <= -5e+153)
		tmp = pi * l_m;
	elseif (t_0 <= -4e-265)
		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_] := Block[{t$95$0 = 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]}, N[(l$95$s * If[LessEqual[t$95$0, -5e+153], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -4e-265], (-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)))) < -5.00000000000000018e153 or -3.99999999999999994e-265 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 69.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. lower-*.f64 N/A

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

      2. lower-PI.f64 79.3

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

   5. Simplified 79.3%

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

   if -5.00000000000000018e153 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -3.99999999999999994e-265

   1. Initial program 96.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-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. lower-*.f64 92.8

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

   5. Simplified 92.8%

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

   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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-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. lower-*.f64 26.4

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

   8. Simplified 26.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -4 \cdot 10^{-265}:\\ \;\;\;\;-\ell \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 85.1

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

   4. Applied egg-rr 85.1%

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

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

   1. Initial program 68.0%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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 88.5%

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

Alternative 5: 98.7% accurate, 1.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 20000000000000.0)
    (+
     (* PI l_m)
     (/
      -1.0
      (*
       F
       (/
        (fma
         (* F (* l_m l_m))
         (/ (* (* PI (* PI PI)) 0.3333333333333333) (* PI (- PI)))
         (/ 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) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F * (fma((F * (l_m * l_m)), (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * 0.3333333333333333) / (((double) M_PI) * -((double) M_PI))), (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) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F * Float64(fma(Float64(F * Float64(l_m * l_m)), Float64(Float64(Float64(pi * Float64(pi * pi)) * 0.3333333333333333) / Float64(pi * Float64(-pi))), Float64(F / 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], 20000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F * N[(N[(N[(F * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(Pi * (-Pi)), $MachinePrecision]), $MachinePrecision] + N[(F / Pi), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 85.1

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

   4. Applied egg-rr 85.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.1

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

   6. Applied egg-rr 85.1%

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

   7. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   9. Simplified 90.3%

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

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

   1. Initial program 68.0%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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 92.5%

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

Alternative 6: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(l\_m \cdot \mathsf{fma}\left(l\_m, l\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{0.3333333333333333}{F}\right), \frac{\pi}{F}\right), \frac{-1}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000000000000.0)
    (fma
     (*
      l_m
      (fma l_m (* l_m (* (* PI (* PI PI)) (/ 0.3333333333333333 F))) (/ PI F)))
     (/ -1.0 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) <= 5000000000000.0) {
		tmp = fma((l_m * fma(l_m, (l_m * ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * (0.3333333333333333 / F))), (((double) M_PI) / F))), (-1.0 / 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) <= 5000000000000.0)
		tmp = fma(Float64(l_m * fma(l_m, Float64(l_m * Float64(Float64(pi * Float64(pi * pi)) * Float64(0.3333333333333333 / F))), Float64(pi / F))), Float64(-1.0 / 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], 5000000000000.0], N[(N[(l$95$m * N[(l$95$m * N[(l$95$m * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / 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) < 5e12

   1. Initial program 80.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

   4. Applied egg-rr 85.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Simplified 70.0%

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

   if 5e12 < (*.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 l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\ell \cdot \mathsf{fma}\left(\ell, \ell \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \frac{0.3333333333333333}{F}\right), \frac{\pi}{F}\right), \frac{-1}{F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{l\_m}{-F}, \frac{\pi}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000000000000.0)
    (fma (/ l_m (- F)) (/ PI 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) <= 5000000000000.0) {
		tmp = fma((l_m / -F), (((double) M_PI) / 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) <= 5000000000000.0)
		tmp = fma(Float64(l_m / Float64(-F)), Float64(pi / 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], 5000000000000.0], N[(N[(l$95$m / (-F)), $MachinePrecision] * N[(Pi / 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) < 5e12

   1. Initial program 80.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 85.4

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

   4. Applied egg-rr 85.4%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 79.6

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

   7. Simplified 79.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

   9. Applied egg-rr 79.6%

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

   if 5e12 < (*.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 l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\ell}{-F}, \frac{\pi}{F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\pi \cdot l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5000000000000.0)
    (- (* 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) <= 5000000000000.0) {
		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) <= 5000000000000.0) {
		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) <= 5000000000000.0:
		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) <= 5000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(pi * l_m) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 5000000000000.0)
		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], 5000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e12

   1. Initial program 80.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   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. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 85.4

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

   4. Applied egg-rr 85.4%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 79.6

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

   7. Simplified 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-frac-neg2 N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lift-neg.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-/.f64 N/A

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

      20. *-commutative N/A

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

   9. Applied egg-rr 79.6%

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

   if 5e12 < (*.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 l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000000000000:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5000000000000.0)
    (* PI (- l_m (/ l_m (* F F))))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000000000000.0) {
		tmp = ((double) M_PI) * (l_m - (l_m / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 5000000000000.0) {
		tmp = Math.PI * (l_m - (l_m / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 5000000000000.0:
		tmp = math.pi * (l_m - (l_m / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 5000000000000.0)
		tmp = Float64(pi * Float64(l_m - Float64(l_m / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 5000000000000.0)
		tmp = pi * (l_m - (l_m / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5000000000000.0], N[(Pi * N[(l$95$m - N[(l$95$m / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e12

   1. Initial program 80.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 85.3

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

   4. Applied egg-rr 85.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.4

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

   6. Applied egg-rr 85.4%

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

   7. Taylor expanded in l around 0

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

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

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 74.4

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

   9. Simplified 74.4%

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

   if 5e12 < (*.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 l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000000000000:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000000000000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 5000000000000.0)
    (* 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) <= 5000000000000.0) {
		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) <= 5000000000000.0) {
		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) <= 5000000000000.0:
		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) <= 5000000000000.0)
		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) <= 5000000000000.0)
		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], 5000000000000.0], 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) < 5e12

   1. Initial program 80.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-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. lower-*.f64 74.3

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

   5. Simplified 74.3%

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

   if 5e12 < (*.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 l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-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 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000000000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% 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 77.0%

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

3. Taylor expanded in l around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 78.3

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

5. Simplified 78.3%

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

6. Final simplification 78.3%

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

Reproduce

herbie shell --seed 2024211 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))