VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 97.7%

Time: 3.4s

Alternatives: 4

Speedup: 1.8×

Specification

Math

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

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: 76.5% accurate, 1.0× speedup

Math

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

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: 97.7% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.75 \cdot 10^{-10}:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\pi \cdot \frac{\left|\ell\right|}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 3.75e-10)
    (- (* PI (fabs l)) (/ (* PI (/ (fabs l) F)) F))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 3.75e-10) {
		tmp = (((double) M_PI) * fabs(l)) - ((((double) M_PI) * (fabs(l) / F)) / F);
	} else {
		tmp = fabs(l) * ((double) M_PI);
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (Math.abs(l) <= 3.75e-10) {
		tmp = (Math.PI * Math.abs(l)) - ((Math.PI * (Math.abs(l) / F)) / F);
	} else {
		tmp = Math.abs(l) * Math.PI;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	tmp = 0
	if math.fabs(l) <= 3.75e-10:
		tmp = (math.pi * math.fabs(l)) - ((math.pi * (math.fabs(l) / F)) / F)
	else:
		tmp = math.fabs(l) * math.pi
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	tmp = 0.0
	if (abs(l) <= 3.75e-10)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(pi * Float64(abs(l) / F)) / F));
	else
		tmp = Float64(abs(l) * pi);
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (abs(l) <= 3.75e-10)
		tmp = (pi * abs(l)) - ((pi * (abs(l) / F)) / F);
	else
		tmp = abs(l) * pi;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 3.75e-10], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * N[(N[Abs[l], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.74999999999999998e-10

   1. Initial program 76.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 82.4

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

   3. Applied rewrites 82.4%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 74.5

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

   6. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if 3.74999999999999998e-10 < l

   1. Initial program 76.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 2: 92.2% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 3.75 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \left|\ell\right|, \frac{\left(-\left|\ell\right|\right) \cdot \pi}{F \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 3.75e-10)
    (fma PI (fabs l) (/ (* (- (fabs l)) PI) (* F F)))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 3.75e-10) {
		tmp = fma(((double) M_PI), fabs(l), ((-fabs(l) * ((double) M_PI)) / (F * F)));
	} else {
		tmp = fabs(l) * ((double) M_PI);
	}
	return copysign(1.0, l) * tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if (abs(l) <= 3.75e-10)
		tmp = fma(pi, abs(l), Float64(Float64(Float64(-abs(l)) * pi) / Float64(F * F)));
	else
		tmp = Float64(abs(l) * pi);
	end
	return Float64(copysign(1.0, l) * tmp)
end

Wolfram

code[F_, l_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 3.75e-10], N[(Pi * N[Abs[l], $MachinePrecision] + N[(N[((-N[Abs[l], $MachinePrecision]) * Pi), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.74999999999999998e-10

   1. Initial program 76.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 40.8%

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

   4. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 28.4

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

   6. Applied rewrites 28.4%

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

   7. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 38.8

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

   9. Applied rewrites 38.8%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. sum-to-mult-rev N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 69.0

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

   11. Applied rewrites 69.0%

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

   if 3.74999999999999998e-10 < l

   1. Initial program 76.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 3: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ t_1 := \pi \cdot \left|\ell\right|\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 - \frac{1}{F \cdot F} \cdot \tan t\_1 \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\frac{-1 \cdot \frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)) (t_1 (* PI (fabs l))))
   (*
    (copysign 1.0 l)
    (if (<= (- t_1 (* (/ 1.0 (* F F)) (tan t_1))) -1e-248)
      (/ (* -1.0 (/ t_0 F)) F)
      t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double t_1 = ((double) M_PI) * fabs(l);
	double tmp;
	if ((t_1 - ((1.0 / (F * F)) * tan(t_1))) <= -1e-248) {
		tmp = (-1.0 * (t_0 / F)) / F;
	} else {
		tmp = t_0;
	}
	return copysign(1.0, l) * tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = Math.abs(l) * Math.PI;
	double t_1 = Math.PI * Math.abs(l);
	double tmp;
	if ((t_1 - ((1.0 / (F * F)) * Math.tan(t_1))) <= -1e-248) {
		tmp = (-1.0 * (t_0 / F)) / F;
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.fabs(l) * math.pi
	t_1 = math.pi * math.fabs(l)
	tmp = 0
	if (t_1 - ((1.0 / (F * F)) * math.tan(t_1))) <= -1e-248:
		tmp = (-1.0 * (t_0 / F)) / F
	else:
		tmp = t_0
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	t_1 = Float64(pi * abs(l))
	tmp = 0.0
	if (Float64(t_1 - Float64(Float64(1.0 / Float64(F * F)) * tan(t_1))) <= -1e-248)
		tmp = Float64(Float64(-1.0 * Float64(t_0 / F)) / F);
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, l) * tmp)
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = abs(l) * pi;
	t_1 = pi * abs(l);
	tmp = 0.0;
	if ((t_1 - ((1.0 / (F * F)) * tan(t_1))) <= -1e-248)
		tmp = (-1.0 * (t_0 / F)) / F;
	else
		tmp = t_0;
	end
	tmp_2 = (sign(l) * abs(1.0)) * tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-248], N[(N[(-1.0 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision], t$95$0]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -9.9999999999999998e-249

   1. Initial program 76.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. sub-to-fraction N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 58.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-to-fraction-rev N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 62.8

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in F around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 26.6

          \[\leadsto \frac{-1 \cdot \frac{\ell \cdot \pi}{F}}{F} \]

   11. Applied rewrites 26.6%

       \[\leadsto \frac{-1 \cdot \color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]

   if -9.9999999999999998e-249 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 76.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.8

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

   4. Applied rewrites 73.8%

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

Alternative 4: 73.8% accurate, 13.6× speedup

Math

\[\ell \cdot \pi \]

FPCore

(FPCore (F l) :precision binary64 (* l PI))

C

double code(double F, double l) {
	return l * ((double) M_PI);
}

Java

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

Python

def code(F, l):
	return l * math.pi

Julia

function code(F, l)
	return Float64(l * pi)
end

MATLAB

function tmp = code(F, l)
	tmp = l * pi;
end

Wolfram

code[F_, l_] := N[(l * Pi), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

2. Taylor expanded in F around inf

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 73.8

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

4. Applied rewrites 73.8%

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

Reproduce

herbie shell --seed 2025175 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))