VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 99.2%

Time: 3.5s

Alternatives: 7

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: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \left|\ell\right|, \frac{\pi \cdot \frac{\left|\ell\right|}{F}}{-F}\right)\\ \mathbf{elif}\;\left|\ell\right| \leq 9200000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\tan t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 1e-9)
      (fma PI (fabs l) (/ (* PI (/ (fabs l) F)) (- F)))
      (if (<= (fabs l) 9200000000000.0)
        (- (* PI (fabs l)) (/ (tan t_0) (* F F)))
        t_0)))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 1e-9) {
		tmp = fma(((double) M_PI), fabs(l), ((((double) M_PI) * (fabs(l) / F)) / -F));
	} else if (fabs(l) <= 9200000000000.0) {
		tmp = (((double) M_PI) * fabs(l)) - (tan(t_0) / (F * F));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, l) * tmp;
}

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 1e-9)
		tmp = fma(pi, abs(l), Float64(Float64(pi * Float64(abs(l) / F)) / Float64(-F)));
	elseif (abs(l) <= 9200000000000.0)
		tmp = Float64(Float64(pi * abs(l)) - Float64(tan(t_0) / Float64(F * F)));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, l) * tmp)
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[Abs[l], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 1e-9], N[(Pi * N[Abs[l], $MachinePrecision] + N[(N[(Pi * N[(N[Abs[l], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[l], $MachinePrecision], 9200000000000.0], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[Tan[t$95$0], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 1.0000000000000001e-9

   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. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 82.6%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 75.2%

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

   6. Applied rewrites 75.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 75.2%

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

   8. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.3%

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

   10. Applied rewrites 75.3%

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

   if 1.0000000000000001e-9 < l < 9.2e12

   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. lower-/.f64 76.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   if 9.2e12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.5%

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

   4. Applied rewrites 73.5%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 9200000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{1}{\frac{F}{\frac{\tan t\_0}{F}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 9200000000000.0)
      (- (* PI (fabs l)) (/ 1.0 (/ F (/ (tan t_0) F))))
      t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 9200000000000.0) {
		tmp = (((double) M_PI) * fabs(l)) - (1.0 / (F / (tan(t_0) / 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 tmp;
	if (Math.abs(l) <= 9200000000000.0) {
		tmp = (Math.PI * Math.abs(l)) - (1.0 / (F / (Math.tan(t_0) / F)));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.fabs(l) * math.pi
	tmp = 0
	if math.fabs(l) <= 9200000000000.0:
		tmp = (math.pi * math.fabs(l)) - (1.0 / (F / (math.tan(t_0) / F)))
	else:
		tmp = t_0
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 9200000000000.0)
		tmp = Float64(Float64(pi * abs(l)) - Float64(1.0 / Float64(F / Float64(tan(t_0) / 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;
	tmp = 0.0;
	if (abs(l) <= 9200000000000.0)
		tmp = (pi * abs(l)) - (1.0 / (F / (tan(t_0) / 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 9200000000000.0], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(F / N[(N[Tan[t$95$0], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.2e12

   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. div-flip N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-/.f64 82.6%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.6%

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

   3. Applied rewrites 82.6%

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

   if 9.2e12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.5%

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

   4. Applied rewrites 73.5%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 9200000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{1}{\frac{F}{\tan t\_0} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 9200000000000.0)
      (- (* PI (fabs l)) (/ 1.0 (* (/ F (tan t_0)) F)))
      t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 9200000000000.0) {
		tmp = (((double) M_PI) * fabs(l)) - (1.0 / ((F / tan(t_0)) * 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 tmp;
	if (Math.abs(l) <= 9200000000000.0) {
		tmp = (Math.PI * Math.abs(l)) - (1.0 / ((F / Math.tan(t_0)) * F));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, l) * tmp;
}

Python

def code(F, l):
	t_0 = math.fabs(l) * math.pi
	tmp = 0
	if math.fabs(l) <= 9200000000000.0:
		tmp = (math.pi * math.fabs(l)) - (1.0 / ((F / math.tan(t_0)) * F))
	else:
		tmp = t_0
	return math.copysign(1.0, l) * tmp

Julia

function code(F, l)
	t_0 = Float64(abs(l) * pi)
	tmp = 0.0
	if (abs(l) <= 9200000000000.0)
		tmp = Float64(Float64(pi * abs(l)) - Float64(1.0 / Float64(Float64(F / tan(t_0)) * 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;
	tmp = 0.0;
	if (abs(l) <= 9200000000000.0)
		tmp = (pi * abs(l)) - (1.0 / ((F / tan(t_0)) * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 9200000000000.0], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[(F / N[Tan[t$95$0], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.2e12

   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. div-flip N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-/.f64 82.6%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.6%

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

   3. Applied rewrites 82.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lower-/.f64 82.6%

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

   5. Applied rewrites 82.6%

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

   if 9.2e12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.5%

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

   4. Applied rewrites 73.5%

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

Alternative 4: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 9200000000000:\\ \;\;\;\;\pi \cdot \left|\ell\right| - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 9200000000000.0)
      (- (* PI (fabs l)) (/ (/ (tan t_0) F) F))
      t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 9200000000000.0) {
		tmp = (((double) M_PI) * fabs(l)) - ((tan(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 tmp;
	if (Math.abs(l) <= 9200000000000.0) {
		tmp = (Math.PI * Math.abs(l)) - ((Math.tan(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
	tmp = 0
	if math.fabs(l) <= 9200000000000.0:
		tmp = (math.pi * math.fabs(l)) - ((math.tan(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)
	tmp = 0.0
	if (abs(l) <= 9200000000000.0)
		tmp = Float64(Float64(pi * abs(l)) - Float64(Float64(tan(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;
	tmp = 0.0;
	if (abs(l) <= 9200000000000.0)
		tmp = (pi * abs(l)) - ((tan(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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 9200000000000.0], N[(N[(Pi * N[Abs[l], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Tan[t$95$0], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.2e12

   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 82.7%

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

      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 82.7%

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

   3. Applied rewrites 82.7%

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

   if 9.2e12 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.5%

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

   4. Applied rewrites 73.5%

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

Alternative 5: 98.4% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 235000000000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \left|\ell\right|, \frac{\pi \cdot \frac{\left|\ell\right|}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\ell\right| \cdot \pi\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (*
  (copysign 1.0 l)
  (if (<= (fabs l) 235000000000.0)
    (fma PI (fabs l) (/ (* PI (/ (fabs l) F)) (- F)))
    (* (fabs l) PI))))

C

double code(double F, double l) {
	double tmp;
	if (fabs(l) <= 235000000000.0) {
		tmp = fma(((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;
}

Julia

function code(F, l)
	tmp = 0.0
	if (abs(l) <= 235000000000.0)
		tmp = fma(pi, abs(l), Float64(Float64(pi * Float64(abs(l) / F)) / Float64(-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], 235000000000.0], N[(Pi * N[Abs[l], $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 < 2.35e11

   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. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 82.6%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 75.2%

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

   6. Applied rewrites 75.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 75.2%

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

   8. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.3%

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

   10. Applied rewrites 75.3%

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

   if 2.35e11 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.5%

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

   4. Applied rewrites 73.5%

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

Alternative 6: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\ell\right| \cdot \pi\\ \mathsf{copysign}\left(1, \ell\right) \cdot \begin{array}{l} \mathbf{if}\;\left|\ell\right| \leq 235000000000:\\ \;\;\;\;t\_0 - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (fabs l) PI)))
   (*
    (copysign 1.0 l)
    (if (<= (fabs l) 235000000000.0) (- t_0 (/ (/ t_0 F) F)) t_0))))

C

double code(double F, double l) {
	double t_0 = fabs(l) * ((double) M_PI);
	double tmp;
	if (fabs(l) <= 235000000000.0) {
		tmp = t_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 tmp;
	if (Math.abs(l) <= 235000000000.0) {
		tmp = t_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
	tmp = 0
	if math.fabs(l) <= 235000000000.0:
		tmp = t_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)
	tmp = 0.0
	if (abs(l) <= 235000000000.0)
		tmp = Float64(t_0 - Float64(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;
	tmp = 0.0;
	if (abs(l) <= 235000000000.0)
		tmp = t_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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[l], $MachinePrecision], 235000000000.0], N[(t$95$0 - N[(N[(t$95$0 / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.35e11

   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. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 82.6%

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

   4. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 75.2%

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

   6. Applied rewrites 75.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 75.2%

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

   8. Applied rewrites 75.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 75.2%

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

   10. Applied rewrites 75.2%

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

   if 2.35e11 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 73.5%

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

   4. Applied rewrites 73.5%

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

Alternative 7: 73.5% 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

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

   1. lower-*.f64 N/A

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

   2. lower-PI.f64 73.5%

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

4. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2025193 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))