VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 99.1%

Time: 17.4s

Alternatives: 5

Speedup: 3.7×

• could not determine a ground truth

  1. F = 7.926842259312124e-247
  2. l = 2968631814910610.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 40000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 40000000000.0)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 40000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 40000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 40000000000.0:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 40000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 40000000000.0)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 40000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e10

   1. Initial program 80.2%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. PI-lowering-PI.f64 86.2

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

   4. Applied egg-rr 86.2%

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

   if 4e10 < (*.f64 (PI.f64) l)

   1. Initial program 62.1%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.7

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 99.7

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

   7. Applied egg-rr 99.7%

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

Alternative 2: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 40000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\pi \cdot l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 40000000000.0)
    (- (* PI l_m) (/ (/ (* PI l_m) F) F))
    (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 40000000000.0) {
		tmp = (((double) M_PI) * l_m) - (((((double) M_PI) * l_m) / F) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 40000000000.0) {
		tmp = (Math.PI * l_m) - (((Math.PI * l_m) / F) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 40000000000.0:
		tmp = (math.pi * l_m) - (((math.pi * l_m) / F) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 40000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(pi * l_m) / F) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 40000000000.0)
		tmp = (pi * l_m) - (((pi * l_m) / F) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 40000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e10

   1. Initial program 80.2%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. PI-lowering-PI.f64 86.2

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

   4. Applied egg-rr 86.2%

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

   5. Taylor expanded in l around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. PI-lowering-PI.f64 80.2

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

   7. Simplified 80.2%

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

   if 4e10 < (*.f64 (PI.f64) l)

   1. Initial program 62.1%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.7

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 99.7

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 40000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 40000000000:\\ \;\;\;\;l\_m \cdot \mathsf{fma}\left(\frac{-1}{F \cdot F}, \pi, \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 40000000000.0)
    (* l_m (fma (/ -1.0 (* F F)) PI PI))
    (* 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) <= 40000000000.0) {
		tmp = l_m * fma((-1.0 / (F * F)), ((double) M_PI), ((double) M_PI));
	} 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) <= 40000000000.0)
		tmp = Float64(l_m * fma(Float64(-1.0 / Float64(F * F)), pi, pi));
	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], 40000000000.0], N[(l$95$m * N[(N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * Pi + Pi), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e10

   1. Initial program 80.2%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. PI-lowering-PI.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 74.2

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

   5. Simplified 74.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-mul-1 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. PI-lowering-PI.f64 74.3

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

   7. Applied egg-rr 74.3%

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

   if 4e10 < (*.f64 (PI.f64) l)

   1. Initial program 62.1%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.7

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 99.7

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

   7. Applied egg-rr 99.7%

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

Alternative 4: 92.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 40000000000:\\ \;\;\;\;l\_m \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 40000000000.0) (* l_m (- PI (/ PI (* F F)))) (* PI l_m))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 40000000000.0) {
		tmp = l_m * (((double) M_PI) - (((double) M_PI) / (F * F)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 40000000000.0) {
		tmp = l_m * (Math.PI - (Math.PI / (F * F)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 40000000000.0:
		tmp = l_m * (math.pi - (math.pi / (F * F)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 40000000000.0)
		tmp = Float64(l_m * Float64(pi - Float64(pi / Float64(F * F))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 40000000000.0)
		tmp = l_m * (pi - (pi / (F * F)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 40000000000.0], N[(l$95$m * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e10

   1. Initial program 80.2%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. PI-lowering-PI.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 74.2

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

   5. Simplified 74.2%

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

   if 4e10 < (*.f64 (PI.f64) l)

   1. Initial program 62.1%

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

   3. Taylor expanded in l around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

      3. PI-lowering-PI.f64 99.7

         \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 99.7

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

   7. Applied egg-rr 99.7%

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

Alternative 5: 74.0% 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 75.2%

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

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right) + 0} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), 0\right)} \]

   3. PI-lowering-PI.f64 77.9

      \[\leadsto \mathsf{fma}\left(\ell, \color{blue}{\pi}, 0\right) \]

5. Simplified 77.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \pi, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 77.9

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

7. Applied egg-rr 77.9%

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

Reproduce

herbie shell --seed 2024197 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))