VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.8% → 99.3%

Time: 4.4s

Alternatives: 11

Speedup: 4.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1950000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 1950000000000.0)
    (- (* 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 (l_m <= 1950000000000.0) {
		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 (l_m <= 1950000000000.0) {
		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 l_m <= 1950000000000.0:
		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 (l_m <= 1950000000000.0)
		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 (l_m <= 1950000000000.0)
		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[l$95$m, 1950000000000.0], 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 l < 1.95e12

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 83.4

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

   4. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-neg.f64 83.4

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

   6. Applied rewrites 83.4%

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

   if 1.95e12 < l

   1. Initial program 53.2%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1950000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+296}:\\ \;\;\;\;\frac{\left(-\pi\right) \cdot l\_m}{F \cdot F}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(F \cdot l\_m\right) \cdot \pi}{F}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

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) (* (/ 1.0 (* F F)) (tan (* PI l_m))))))
   (*
    l_s
    (if (<= t_0 -2e+296)
      (/ (* (- PI) l_m) (* F F))
      (if (<= t_0 -5e+143)
        (/ (* (* F l_m) PI) F)
        (if (<= t_0 -2e-231) (* (- 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) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)));
	double tmp;
	if (t_0 <= -2e+296) {
		tmp = (-((double) M_PI) * l_m) / (F * F);
	} else if (t_0 <= -5e+143) {
		tmp = ((F * l_m) * ((double) M_PI)) / F;
	} else if (t_0 <= -2e-231) {
		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) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)));
	double tmp;
	if (t_0 <= -2e+296) {
		tmp = (-Math.PI * l_m) / (F * F);
	} else if (t_0 <= -5e+143) {
		tmp = ((F * l_m) * Math.PI) / F;
	} else if (t_0 <= -2e-231) {
		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) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))
	tmp = 0
	if t_0 <= -2e+296:
		tmp = (-math.pi * l_m) / (F * F)
	elif t_0 <= -5e+143:
		tmp = ((F * l_m) * math.pi) / F
	elif t_0 <= -2e-231:
		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(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m))))
	tmp = 0.0
	if (t_0 <= -2e+296)
		tmp = Float64(Float64(Float64(-pi) * l_m) / Float64(F * F));
	elseif (t_0 <= -5e+143)
		tmp = Float64(Float64(Float64(F * l_m) * pi) / F);
	elseif (t_0 <= -2e-231)
		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) - ((1.0 / (F * F)) * tan((pi * l_m)));
	tmp = 0.0;
	if (t_0 <= -2e+296)
		tmp = (-pi * l_m) / (F * F);
	elseif (t_0 <= -5e+143)
		tmp = ((F * l_m) * pi) / F;
	elseif (t_0 <= -2e-231)
		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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[t$95$0, -2e+296], N[(N[((-Pi) * l$95$m), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e+143], N[(N[(N[(F * l$95$m), $MachinePrecision] * Pi), $MachinePrecision] / F), $MachinePrecision], If[LessEqual[t$95$0, -2e-231], N[((-l$95$m) * N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.99999999999999996e296

   1. Initial program 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 52.8

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in F around 0

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

      1. frac-times N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. frac-times N/A

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

      8. quot-tan N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

   7. Applied rewrites 33.9%

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

   8. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-neg.f64 27.5

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

   10. Applied rewrites 27.5%

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

   if -1.99999999999999996e296 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -5.00000000000000012e143

   1. Initial program 73.5%

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

   3. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 28.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 30.4%

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

   8. Taylor expanded in F around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 63.3

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

   10. Applied rewrites 63.3%

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

   if -5.00000000000000012e143 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -2e-231

   1. Initial program 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in F around 0

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

      1. frac-times N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. frac-times N/A

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

      8. quot-tan N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

         \[\leadsto -\ell \cdot \frac{\pi}{{F}^{2}} \]

      7. pow2 N/A

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

      8. lift-*.f64 34.4

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

   10. Applied rewrites 34.4%

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

   if -2e-231 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 70.1%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 73.1

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\pi \cdot \ell} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \leq -2 \cdot 10^{+296}:\\ \;\;\;\;\frac{\left(-\pi\right) \cdot \ell}{F \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(F \cdot \ell\right) \cdot \pi}{F}\\ \mathbf{elif}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right) \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m)))) -2e-231)
    (* (- l_m) (/ PI (* F F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (((((double) M_PI) * l_m) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)))) <= -2e-231) {
		tmp = -l_m * (((double) M_PI) / (F * F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (((Math.PI * l_m) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)))) <= -2e-231) {
		tmp = -l_m * (Math.PI / (F * F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if ((math.pi * l_m) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))) <= -2e-231:
		tmp = -l_m * (math.pi / (F * F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m)))) <= -2e-231)
		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)
	tmp = 0.0;
	if (((pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)))) <= -2e-231)
		tmp = -l_m * (pi / (F * F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-231], 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)))) < -2e-231

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in F around 0

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

      1. frac-times N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. associate-*r/ N/A

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

      7. frac-times N/A

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

      8. quot-tan N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

   7. Applied rewrites 28.9%

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

   8. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

         \[\leadsto -\ell \cdot \frac{\pi}{{F}^{2}} \]

      7. pow2 N/A

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

      8. lift-*.f64 23.4

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

   10. Applied rewrites 23.4%

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

   if -2e-231 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 70.1%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 73.1

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 73.1%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1950000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot l\_m\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 1950000000000.0)
    (- (* PI l_m) (* (/ 1.0 F) (/ (tan (* PI l_m)) F)))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 1950000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((1.0 / F) * (tan((((double) M_PI) * l_m)) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 1950000000000.0) {
		tmp = (Math.PI * l_m) - ((1.0 / F) * (Math.tan((Math.PI * l_m)) / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 1950000000000.0:
		tmp = (math.pi * l_m) - ((1.0 / F) * (math.tan((math.pi * l_m)) / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 1950000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / F) * Float64(tan(Float64(pi * l_m)) / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 1950000000000.0)
		tmp = (pi * l_m) - ((1.0 / F) * (tan((pi * l_m)) / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 1950000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] * N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.95e12

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 83.4

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

   4. Applied rewrites 83.4%

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

   if 1.95e12 < l

   1. Initial program 53.2%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1950000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1950000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{1}{F} \cdot \tan \left(\pi \cdot l\_m\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 1950000000000.0)
    (- (* PI l_m) (/ (* (/ 1.0 F) (tan (* PI l_m))) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 1950000000000.0) {
		tmp = (((double) M_PI) * l_m) - (((1.0 / F) * tan((((double) M_PI) * l_m))) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 1950000000000.0) {
		tmp = (Math.PI * l_m) - (((1.0 / F) * Math.tan((Math.PI * l_m))) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 1950000000000.0:
		tmp = (math.pi * l_m) - (((1.0 / F) * math.tan((math.pi * l_m))) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 1950000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(1.0 / F) * tan(Float64(pi * l_m))) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 1950000000000.0)
		tmp = (pi * l_m) - (((1.0 / F) * tan((pi * l_m))) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 1950000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(1.0 / F), $MachinePrecision] * N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.95e12

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 83.4

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

   4. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-tan.f64 N/A

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

      18. lift-neg.f64 83.4

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

   6. Applied rewrites 83.4%

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

   if 1.95e12 < l

   1. Initial program 53.2%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1950000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.16 \cdot 10^{-40}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{\pi \cdot l\_m}{-F}}{F}\\ \mathbf{elif}\;l\_m \leq 1950000000000:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{\tan \left(\pi \cdot l\_m\right)}{\left(-F\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 1.16e-40)
    (+ (* PI l_m) (/ (/ (* PI l_m) (- F)) F))
    (if (<= l_m 1950000000000.0)
      (fma 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 (l_m <= 1.16e-40) {
		tmp = (((double) M_PI) * l_m) + (((((double) M_PI) * l_m) / -F) / F);
	} else if (l_m <= 1950000000000.0) {
		tmp = fma(((double) M_PI), l_m, (tan((((double) M_PI) * l_m)) / (-F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 1.16e-40)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(pi * l_m) / Float64(-F)) / F));
	elseif (l_m <= 1950000000000.0)
		tmp = fma(pi, l_m, Float64(tan(Float64(pi * l_m)) / Float64(Float64(-F) * F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 1.16e-40], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(Pi * l$95$m), $MachinePrecision] / (-F)), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1950000000000.0], N[(Pi * l$95$m + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[((-F) * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.15999999999999991e-40

   1. Initial program 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-neg.f64 82.1

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

   6. Applied rewrites 82.1%

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

   7. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-/.f64 76.0

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

   9. Applied rewrites 76.0%

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

   if 1.15999999999999991e-40 < l < 1.95e12

   1. Initial program 99.8%

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

   3. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\pi \cdot \ell\right) + \pi \cdot \ell \]

      5. lift-tan.f64 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\pi \cdot \ell\right) + \pi \cdot \ell \]

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. associate-*r/ N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 99.8%

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

   if 1.95e12 < l

   1. Initial program 53.2%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.16 \cdot 10^{-40}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{\pi \cdot \ell}{-F}}{F}\\ \mathbf{elif}\;\ell \leq 1950000000000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\tan \left(\pi \cdot \ell\right)}{\left(-F\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 160000:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{\pi \cdot l\_m}{-F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 160000.0) (+ (* 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 (l_m <= 160000.0) {
		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 (l_m <= 160000.0) {
		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 l_m <= 160000.0:
		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 (l_m <= 160000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(pi * l_m) / Float64(-F)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 160000.0)
		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[l$95$m, 160000.0], 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 l < 1.6e5

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 83.3

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

   4. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. lift-neg.f64 83.4

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

   6. Applied rewrites 83.4%

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

   7. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-/.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 1.6e5 < l

   1. Initial program 54.0%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 160000:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{\pi \cdot \ell}{-F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 160000:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{\left(-l\_m\right) \cdot \pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 160000.0) (fma PI l_m (/ (* (- l_m) PI) (* F F))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 160000.0) {
		tmp = fma(((double) M_PI), l_m, ((-l_m * ((double) M_PI)) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 160000.0)
		tmp = fma(pi, l_m, Float64(Float64(Float64(-l_m) * pi) / Float64(F * F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 160000.0], N[(Pi * l$95$m + N[(N[((-l$95$m) * Pi), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.6e5

   1. Initial program 73.1%

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

   3. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\pi \cdot \ell\right) + \pi \cdot \ell \]

      5. lift-tan.f64 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\pi \cdot \ell\right) + \pi \cdot \ell \]

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. associate-*r/ N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-neg.f64 68.8

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 68.8

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

   10. Applied rewrites 68.8%

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

   if 1.6e5 < l

   1. Initial program 54.0%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 160000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\left(-\ell\right) \cdot \pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 160000:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 160000.0) (* (- PI (/ PI (* F F))) 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 (l_m <= 160000.0) {
		tmp = (((double) M_PI) - (((double) M_PI) / (F * F))) * 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 (l_m <= 160000.0) {
		tmp = (Math.PI - (Math.PI / (F * F))) * 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 l_m <= 160000.0:
		tmp = (math.pi - (math.pi / (F * F))) * 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 (l_m <= 160000.0)
		tmp = Float64(Float64(pi - Float64(pi / Float64(F * F))) * 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 (l_m <= 160000.0)
		tmp = (pi - (pi / (F * F))) * 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[l$95$m, 160000.0], N[(N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.6e5

   1. Initial program 73.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

         \[\leadsto \left(\pi - \frac{\pi}{{F}^{2}}\right) \cdot \ell \]

      7. pow2 N/A

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

      8. lift-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.6e5 < l

   1. Initial program 54.0%

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

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

         \[\leadsto \pi \cdot \ell \]

   5. Applied rewrites 99.6%

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

Alternative 10: 73.7% 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 68.9%

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

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 69.8

      \[\leadsto \pi \cdot \ell \]

5. Applied rewrites 69.8%

   \[\leadsto \color{blue}{\pi \cdot \ell} \]
6. Add Preprocessing

Alternative 11: 3.1% accurate, 135.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot 0 \end{array} \]

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 0.0))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * 0.0;
}

Fortran

l\_m =     private
l\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_s, f, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_s
    real(8), intent (in) :: f
    real(8), intent (in) :: l_m
    code = l_s * 0.0d0
end function

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 * 0.0;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * 0.0

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * 0.0)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 68.9%

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

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. lower-tan.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 50.1

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in l around 0

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{\sin \mathsf{PI}\left(\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)} \]

   2. lift-PI.f64 N/A

      \[\leadsto -1 \cdot \frac{\sin \mathsf{PI}\left(\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)} \]

   3. tan-+PI-rev N/A

      \[\leadsto -1 \cdot \frac{\sin \mathsf{PI}\left(\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)} \]

   4. lift-PI.f64 N/A

      \[\leadsto -1 \cdot \frac{\sin \mathsf{PI}\left(\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto -1 \cdot \frac{\sin \mathsf{PI}\left(\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)} \]

   6. associate-*r/ N/A

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

   7. sin-PI N/A

      \[\leadsto \frac{-1 \cdot 0}{{F}^{\color{blue}{2}} \cdot \cos \mathsf{PI}\left(\right)} \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. sin-PI N/A

       \[\leadsto \frac{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\sin \mathsf{PI}\left(\right)}\right)}{{F}^{2} \cdot \cos \mathsf{PI}\left(\right)} \]

   12. log-pow-rev N/A

       \[\leadsto \frac{\sin \mathsf{PI}\left(\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)}{\color{blue}{{F}^{2}} \cdot \cos \mathsf{PI}\left(\right)} \]

   13. add-log-exp N/A

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

   14. *-commutative N/A

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

7. Applied rewrites 2.4%

   \[\leadsto \color{blue}{\frac{0}{F \cdot F}} \]

8. Taylor expanded in F around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 2.9%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))