VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.3%

Time: 16.5s

Alternatives: 8

Speedup: 3.7×

• could not determine a ground truth

  1. F = 7.3579658613052665e-292
  2. l = -3727329285076699.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.7

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

   4. Applied egg-rr 88.7%

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

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

   1. Initial program 63.7%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]
   6. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied egg-rr 99.7%

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

Alternative 2: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, -\frac{\frac{\pi \cdot l\_m}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e-8)
    (fma PI l_m (- (/ (/ (* PI l_m) F) F)))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = fma(((double) M_PI), l_m, -(((((double) M_PI) * l_m) / F) / F));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-8)
		tmp = fma(pi, l_m, Float64(-Float64(Float64(Float64(pi * l_m) / F) / F)));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * l$95$m + (-N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision])), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 77.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 72.5

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

   7. Simplified 72.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 82.5

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

   9. Applied egg-rr 82.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]
   6. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, -\frac{\frac{\pi \cdot \ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l\_m, -\frac{\frac{\pi \cdot l\_m}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e-8)
    (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 ((((double) M_PI) * l_m) <= 1e-8) {
		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 (Float64(pi * l_m) <= 1e-8)
		tmp = fma(pi, l_m, Float64(-Float64(Float64(Float64(pi * l_m) / F) / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * l$95$m + (-N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision])), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 77.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 72.5

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

   7. Simplified 72.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 82.5

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

   9. Applied egg-rr 82.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, -\frac{\frac{\pi \cdot \ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\pi \cdot l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e-8) (- (* PI l_m) (/ (/ (* PI l_m) F) F)) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = (((double) M_PI) * l_m) - (((((double) M_PI) * l_m) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e-8) {
		tmp = (Math.PI * l_m) - (((Math.PI * l_m) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e-8:
		tmp = (math.pi * l_m) - (((math.pi * l_m) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(pi * l_m) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1e-8)
		tmp = (pi * l_m) - (((pi * l_m) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.6

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

   4. Applied egg-rr 88.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 82.5

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

   7. Simplified 82.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \frac{-l\_m}{F \cdot F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e-8)
    (fma PI (/ (- l_m) (* F F)) (* PI l_m))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = fma(((double) M_PI), (-l_m / (F * F)), (((double) M_PI) * l_m));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-8)
		tmp = fma(pi, Float64(Float64(-l_m) / Float64(F * F)), Float64(pi * l_m));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * N[((-l$95$m) / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 77.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 72.5

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

   7. Simplified 72.5%

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

   9. Applied egg-rr 72.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \frac{-\ell}{F \cdot F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(l\_m - \frac{l\_m}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= (* PI l_m) 1e-8) (* PI (- l_m (/ l_m (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = ((double) M_PI) * (l_m - (l_m / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e-8) {
		tmp = Math.PI * (l_m - (l_m / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e-8:
		tmp = math.pi * (l_m - (l_m / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(pi * Float64(l_m - Float64(l_m / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1e-8)
		tmp = pi * (l_m - (l_m / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(Pi * N[(l$95$m - N[(l$95$m / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 77.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 72.5

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

   7. Simplified 72.5%

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

   9. Applied egg-rr 72.5%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-PI.f64 N/A

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

       6. distribute-lft-out N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 72.5

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

       11. lift-/.f64 N/A

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

       12. lift-neg.f64 N/A

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

       13. distribute-frac-neg2 N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-/.f64 72.5

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

   11. Applied egg-rr 72.5%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-8}:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= (* PI l_m) 1e-8) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e-8) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e-8) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e-8:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e-8)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1e-8)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e-8], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 77.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 77.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 72.5

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

   7. Simplified 72.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. unsub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 71.7

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

   9. Applied egg-rr 71.7%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 64.9%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 22.5× speedup

Math

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

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 74.8%

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

3. Taylor expanded in l around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 71.7

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

5. Simplified 71.7%

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

6. Final simplification 71.7%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))