VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 98.5%

Time: 3.3s

Alternatives: 7

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: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 16:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 16.0) (- (* 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 <= 16.0) {
		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 <= 16.0) {
		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 <= 16.0:
		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 <= 16.0)
		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 <= 16.0)
		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, 16.0], 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 < 16

   1. Initial program 88.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. 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 99.5

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

   3. Applied rewrites 99.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg-rev N/A

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

      11. *-lft-identity N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-tan.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 16 < l

   1. Initial program 64.7%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 97.4

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

   4. Applied rewrites 97.4%

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

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\frac{-\pi}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

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 -5e+162)
      (* PI l_m)
      (if (<= t_0 -1e-252) (* (/ (- 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 t_0 = (((double) M_PI) * l_m) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)));
	double tmp;
	if (t_0 <= -5e+162) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -1e-252) {
		tmp = (-((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 t_0 = (Math.PI * l_m) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)));
	double tmp;
	if (t_0 <= -5e+162) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -1e-252) {
		tmp = (-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):
	t_0 = (math.pi * l_m) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))
	tmp = 0
	if t_0 <= -5e+162:
		tmp = math.pi * l_m
	elif t_0 <= -1e-252:
		tmp = (-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)
	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 <= -5e+162)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -1e-252)
		tmp = Float64(Float64(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)
	t_0 = (pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)));
	tmp = 0.0;
	if (t_0 <= -5e+162)
		tmp = pi * l_m;
	elseif (t_0 <= -1e-252)
		tmp = (-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_] := 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, -5e+162], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -1e-252], N[(N[((-Pi) / N[(F * F), $MachinePrecision]), $MachinePrecision] * l$95$m), $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)))) < -4.9999999999999997e162 or -9.99999999999999943e-253 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 74.4%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 82.5

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

   4. Applied rewrites 82.5%

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

   if -4.9999999999999997e162 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -9.99999999999999943e-253

   1. Initial program 92.8%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   3. 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 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.6

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

   7. Applied rewrites 89.6%

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

Alternative 3: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\frac{-\pi \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (- (* PI l_m) (* (/ 1.0 (* F F)) (tan (* PI l_m))))))
   (*
    l_s
    (if (<= t_0 -5e+162)
      (* PI l_m)
      (if (<= t_0 -1e-252) (/ (- (* PI l_m)) (* F F)) (* PI l_m))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = (((double) M_PI) * l_m) - ((1.0 / (F * F)) * tan((((double) M_PI) * l_m)));
	double tmp;
	if (t_0 <= -5e+162) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -1e-252) {
		tmp = -(((double) M_PI) * l_m) / (F * F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double t_0 = (Math.PI * l_m) - ((1.0 / (F * F)) * Math.tan((Math.PI * l_m)));
	double tmp;
	if (t_0 <= -5e+162) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -1e-252) {
		tmp = -(Math.PI * l_m) / (F * F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = (math.pi * l_m) - ((1.0 / (F * F)) * math.tan((math.pi * l_m)))
	tmp = 0
	if t_0 <= -5e+162:
		tmp = math.pi * l_m
	elif t_0 <= -1e-252:
		tmp = -(math.pi * l_m) / (F * F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(Float64(pi * l_m) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l_m))))
	tmp = 0.0
	if (t_0 <= -5e+162)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -1e-252)
		tmp = 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)
	t_0 = (pi * l_m) - ((1.0 / (F * F)) * tan((pi * l_m)));
	tmp = 0.0;
	if (t_0 <= -5e+162)
		tmp = pi * l_m;
	elseif (t_0 <= -1e-252)
		tmp = -(pi * l_m) / (F * F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := Block[{t$95$0 = N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(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, -5e+162], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -1e-252], N[((-N[(Pi * l$95$m), $MachinePrecision]) / N[(F * F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -4.9999999999999997e162 or -9.99999999999999943e-253 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 74.4%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 82.5

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

   4. Applied rewrites 82.5%

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

   if -4.9999999999999997e162 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -9.99999999999999943e-253

   1. Initial program 92.8%

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

   2. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 89.7

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

   7. Applied rewrites 89.7%

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

Alternative 4: 98.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m}{F} \cdot \frac{\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 0.5) (- (* PI l_m) (* (/ l_m F) (/ PI 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 <= 0.5) {
		tmp = (((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / 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 <= 0.5) {
		tmp = (Math.PI * l_m) - ((l_m / F) * (Math.PI / 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 <= 0.5:
		tmp = (math.pi * l_m) - ((l_m / F) * (math.pi / 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 <= 0.5)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / 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 <= 0.5)
		tmp = (pi * l_m) - ((l_m / F) * (pi / 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, 0.5], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 88.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. 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 99.5

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

   3. Applied rewrites 99.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg-rev N/A

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

      11. *-lft-identity N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-tan.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in l around 0

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

      1. pow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 0.5 < l

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 5: 92.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, \frac{-\pi \cdot l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 0.5) (fma 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 <= 0.5) {
		tmp = fma(((double) M_PI), l_m, (-(((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 <= 0.5)
		tmp = fma(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

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, 0.5], N[(Pi * l$95$m + N[((-N[(Pi * l$95$m), $MachinePrecision]) / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 88.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   3. 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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in F around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 0.5 < l

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 6: 92.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 0.5) (* (- 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 <= 0.5) {
		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 <= 0.5) {
		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 <= 0.5:
		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 <= 0.5)
		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 <= 0.5)
		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, 0.5], 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 < 0.5

   1. Initial program 88.1%

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

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   3. 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 87.7

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

   4. Applied rewrites 87.7%

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

   if 0.5 < l

   1. Initial program 64.9%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   3. 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 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 7: 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 76.6%

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

2. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
3. 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 73.7

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

4. Applied rewrites 73.7%

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

Reproduce

herbie shell --seed 2025096 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))